A Study of Extra Special P-Group 1 2 MURTADHA ALI SHABEEB , DR

ISSN 2319-8885 Vol.02,Issue.19, December-2013, Pages:2223-2234 www.semargroups.org, www.ijsetr.com

A Study of Extra Special P-Group 1 2 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA 1MSc, Dept of Mathematics, SHITAS, Allahabad, UP-INDIA, Email: [email protected]. 2Asst Prof, Dept of Mathematics, SHIATS, Allahabad, UP-INDIA.

Abstract: In this dissertation we have discussed extra special -group. A finite nonabelian -group is called extra special - group if its center is exactly equal to its commutator subgroup. Here we have discussed extra special -groups and we have find that every non-abelian group of order is extra special -group. In particular if then we have two extra special -groups one is dihedral group and another is Hamiltonian group . Here we have also discussed that if is non-abelian group of order , then has order . We have thoroughly discussed the following theorem: Let be a finite extra special group. Then it is central product of non-abelian groups of order . In particular is of order for some . To prove above theorem we have gone through solvability and nilpotency in groups, Frattini subgroups, and different type of bilinear forms.

Keywords: P-Group, Hamiltonian Group.

I. INTRODUCTION only if the structure ( namely the Galois group) possess a Algebra is one of the broad parts of Mathematics, property ( called the solvability ). In the second half of together with number theory, geometry and analysis. For the nineteenth century the notion of the congruences of historical reasons, the word “algebra” has several related Geometric objects was generalized. The development meaning in mathematics, as a single word or with qualifies. during this period was influenced by the work of Lie, The word algebra originated from the title of the book “al- Klein, Poincare and Dehn. The importance of the study of Kitab al-mukhtasar fi hisab al-jabr w'al-muqabala ”, a book permutation Groups, Continuous Groups, Groups of written during the ninth century the Arabian mathematician homeomorphisms and Fundamental Groups was realized named Al-Khwarizmi. The original title was translated as and this lead to the formulation of an Abstract Group. The the science of restoration and reduction, basically meaning notion of an abstract group is present in the works of Caley transposing and combining similar terms of equations. and Vondyck. Translated in Latin to al-jabr (the union of broken parts). led to the term we now refer to as algebra, Algebra was The theory of groups developed slowly but steadily in brought from ancient Babylon, Egypt and India to Europe the first half of the twentieth century with some very via Italy by the Arabs. One of the most fundamental significant contributions by Burnside, Schreier, P.Hall and concept in Mathematics today is that of a group. Germs of others. Theory of finite groups picked up momentum with group was present , even in ancient times , in the study of the works of Brauer and his students in 1955. Theory of congruences of Geometric figures and also in the study of groups, now, has tremendous applications and interest in motions in space .It started taking shape in the beginning itself. There are four major sources in the evolution of of the nineteenth century . One of the most challenging group theory. They are: problem at this time was the problem of solvability of general polynomial equations of degree, ≥ by field and 1. Classical algebra. (J.L. Lagrange, 1770). 2. Number theory (C.F. Gauss, 1801). radical operations (addition, subtraction, multiplication, 3. Geometry (F.Klein, 1874). division by non-zero elements and taking roots for 4. Analysis (S.Lie, 1874, H. Poincare and F. Klein, different ). Abel and Ruffini proved, using the structure 1876). of the set of permutations on the set of roots of the polynomials that a general degree equation ≥ is A. Objectives not solvable by field and radical operations. Galois • To study extra special group. motivated by the work of Abel , attached , to every • To find extra special groups of different polynomial equation a structure (called the Galois group of the polynomial equation ) and proved that a polynomial orders. equation is solvable by field and radical operations if and

B. Commutators Remark: The commutator subgroup is the smallest normal Definition 1: Let be a group. An element of the form, subgroup of by which if we factor we get an abelian which is denoted by ( , ) is called a group. The quotient group is the largest quotient commutator. The subgroup of generated by all group of which is abelian. The group is called the commutators of is called the commutator Subgroup or abelianizer of and is denoted by the derived subgroup of and denoted by [ , . Thus =[ , = ( , ) | , . Definition 3: If is nonabelian simple group then . A group is said to be perfect if its commutator Definition 2: Let and be subsets of a group .The subgroup is itself. Thus a nonabelian simple group is subgroup generated by {( , ) | , } is denoted always perfect. by [ , . Example 1: = : is abelian group. Proposition: Let be a group and , and be elements Hence . Since is nonabelian, and of . since has no nontrivial proper subgroup = .

1. ( , )= if and only if and commute. Example 2: = : is abelian and 2. ( , )= . therefore . The subgroups of which are 3. =( , ) . normal in are , and the trivial group. Since is 4. [ , =[ , . nonabelian it's commutator subgroup is nontrivial. Further 5. ( , )= ( , ). since is nonabelian. Thus = . 6. ( , )= . 7. = = . Where means . Example 3: For , : Since is abelian, . Further for , is simple. From (above proposition 1), we understand that Thus has no nontrivial proper normal subgroups and commutators measure, so to speak, how non-abelian a since is nonabelian, . group is. When the set of commutators consists of 1 only, then the group is abelian. Rather sloppily, the more Proposition: Let be a surjective homomorphism from nonidentity commutators a group has, the more elements of to . Then = . fail to commute with other elements of , and the more non-abelian is. Proof: Since (( , ))= ( )= =( , ), it follows that . Further let Corollary: [ , . ( , ) be a commutator in . Since is surjective, there Proof: To prove , let ( , ) and . exist , such that = , = . Then ( , ) ( , ) = ( ) =( =( , )= (( , )) . Thus all commutators ( ( ( . Just inserted of are in and so . between each elements. Now note that ( Corollary: Let . Then = . = .Similarly ( . So we have ( ) = Proof: The quotient map from to is a surjective ( ( ( . homomorphism and so = { | }= { | , }= . Theorem 1: Let be a normal subgroup of . Then is abelian if and only if . Also if is any C. Solvable Groups subgroup of containing , then it is normal in Definition1: Let be a group. Define subgroups of inductively as follows: Define . Assuming that Proof: is abelian for all has already been defined, define [ , . , = for all Thus ...... of , for all , . This series is called the Commutator Series or the ( , ) for all , ( , ) | , is a Derrived Series of and is called the term of the subgroup of is a subgroup of . derived series.

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 A Study of Extra Special P-Group

Definition 2: A group is said to be Solvable (or Soluble) Example 6: For , is not solvable ⊳ ⊳{ if the derived series of terminates to after finitely and ⊳{ are the only normal series and none of them many steps. The smallest such that = is called the are with abelian factors. derived length of . Proposition: Quotient group of a solvable group is solvable. Example 1: The derived series of is: , , . is solvable of derived length . Proposition: Let be a group and is a solvable normal subgroup of such that is solvable. Then is also Example 2: The derived series of is = , = , solvable. = , = . is solvable of derived length . Proof: Since is solvable it has a normal series Example 3: is nonabelian simple group, thus . . . . Since is . The derived series of is = for all . solvable it has a normal series Thus insolvable. . . . for some , . Since , so that Remark: Every abelian group is solvable. Since = , is a subgroup of . Thus is a subgroup of then the derived series of is of length . . Therefore .

Proposition: A Subgroup of solvable group is solvable. Proposition: Let be a group and , solvable subgroups of . Suppose that . Then is a Proof: Let be a solvable group, a subgroup of solvable subgroup. . . . Suppose . Thus . Proof: Since , and so is a subgroup of . Since is solvable, is solvable. Then Proposition: Homomorphic image of a solvable group is solvable. is isomorphic to . Hence and both are solvable. Thus is solvable. Proof: Suppose that is a solvable group and : → is Proposition: A maximal solvable normal subgroup of a surjective homomrphism . = [ , = ( , ) | , group (if exists) is largest solvable normal subgroup. . (( , ))= ( )= =( , ). Therefore . Proof: If is a maximal solvable normal subgroup and Thus . a solvable normal subgroup, then is also solvable normal subgroup. Since supposed to be maximal, Proposition: A group is solvable if and only if it has a and so . normal series with abelian factors. Corollary: A solvable group is simple if and only if it is a Proof: If is solvable, then the derived series cyclic group of prime order. . . . is a normal series with abelian factors. Conversely suppose that has Proof: Assume that is prime cyclic, then it is abelian. So a normal series = ⊵ ⊵... ⊵ = with abelian = . Hence the derived series of is factors, i.e. is abelian . We show, by . So is solvable. Conversely, let be a induction, that . . solvable which is simple. Then = is a subgroup of is abelian, then . Suppose inductively . . Since is simple, it has no normal subgroup. Thus is abelian, then . Thus = so is abelian. Hence it is prime cyclic group. . Then . Hence is solvable. Example 7: Every group of order where and are prime is solvable: If , then | = and so it is Example 4: is solvable for ⊳ ⊳{ is a normal abelian. Hence it is solvable. Suppose that and series of with abelian factors. . Then the of is normal. Then and (prime cyclic) are solvable. Example 5: is solvable for ⊳ ⊳ ⊳{ is a Hence is solvable. normal series of with abelian factors.

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA

Example 8: Every group of prime power order is solvable: proof: Suppose that = . We show, by induction Let | = . The proof is by induction on . If , on , that ⊆ ∀ . For =0, then is prime cyclic and so solvable. Assume that the = = = and so the result is true for 0. result is true for all those groups of orders , . Assume that ⊆ . We show that Since is prime power order . Hence ⊆ . Let ∈ . Since =[ , | for some . By induction ], ∈ ⊆ ∀ ∈ . This hypothesis is solvable. Since is abelian, shows that = ∀ ∈ .Thus thus is solvable . belongs to the center of . By the definition ∈ . Putting we find that Example 9: Every group of order where , and . Conversely suppose that = . We show, by are distinct primes is solvable: The induction, that ⊆ ∀ . For =0, = of is normal which (prime cyclic) is solvable. Also and so the result is true for =0. Assume that | = and so is also solvable. Hence is solvable. ⊆ . We show that ⊆ . From the definition of , it suffices to show that D. Nilpotent Groups ∈ ∀ ∈ and ∈ . Let ∈ Definition 1: Let be a group. Define subgroups . Since is assumed to be contained in inductively as follows: = , =[ , ] , ∈ . But, then by the definition of =[ , ]= . Assuming that has already been , belongs to the center of . defined. Define =[ , ]. = , Thus = =[ , ]=[ , ] = . Assume for all ∈ .This means that ∈ . Since if and are normal in , then [ , ] is also ∈ . This completes the proof of the fact that normal in . So =[ , ] and ⊆ . Putting , we get that . Therefore by induction hypothesis this is true = . for all . This gives a descending series = ⊵ ⊵ ⊵... ⊵ ⊵ ..of called the Lower Definition 3: A group is said to be nilpotent if Central Series of . = = , =[ , ]= . = or equivalently = for some . A Assume . =[ , ]. Since group is said to be nilpotent of class if = but or equivalently = but . and . Then [ , ] ≠ ≠ [ , ]. Thus . Therefore by Proposition: Every nilpotent group is solvable. induction hypothesis this is true for all . Proof: Let be a nilpotent group. Therefore the lower Definition 2: Define normal subgroups of central series must terminates to at the step, inductively as follows: Define normal subgroup i.e. = . Then = ⊵ ⊵ ⊵... ⊵ = , = the center of . Observe that = . Since . Thus = . Hence = the center of . . . . . Which Supposing that has already been defined, define shows that is solvable. by the equation = . Thus we get an ascending series = ⊴ ⊴ Remark: A solvable group need not be nilpotent. For ⊴... ⊴ ⊴ of normal subgroups of . This example in , , [ , ]= , series is called the Upper Central Series of . [ , ]= . Therefore lower central series not terminates to . Example 1: , = , = , = = }= = = Proposition: A Subgroup of nilpotent group is nilpotent. . Proof: Let be a subgroup of nilpotent group . Since Theorem 1: The lower central series of terminates to . Thus . Assume that . at the step if and only if the upper central series =[ , ] [ , ] . Thus . terminates to at the step ( i.e. = if and only Proposition: Homomorphic image of a nilpotent group is if = ). nilpotent.

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 A Study of Extra Special P-Group proof: Suppose be a surjective =[ , ]= (fro homomorphism. i.e. = and is a nilpotent group m above). . So by induction .Therefore lower central series = ⊵ ⊵ hypothesis, = ∀ . Since ⊵ . . . ⊵ = is terminate at step. and are nilpotent. So suppose = and = = = . Assume that = . Let = . Then = = . = = = = . = = . Therefore by induction hypothesis = ∀ . Since = , Proposition Let be a nontrivial nilpotent group. Then = = . So is also nilpotent. . Thus in a nontrivial nilpotent group there is an element different from identity which commute with Proposition: Quotient group of a nilpotent is nilpotent . each elements of the group. proof: Let be a nilpotent group and ⊴ . Let , Proof: Suppose that . Then = ∈ . . Since = = = = = , = . Proceeding [ , ]=< . . │ ∈ inductively, we see that for all . Hence >=< . .│ ∈ >=< │ can never be and so is not nilpotent. ∈ >= (∵ ∈ ). ( ) = = . Assume that ( )= . Remark: A solvable group need not be nilpotent: is ( )=[ , ( )=[ , ]= solvable but, since , it is not nilpotent . (from above)= = . Proposition: Let be a normal subgroup of a group Therefore by induction hypothesis, = contained in the center of such that is . Since is a nilpotent group so at step, nilpotent. Then is nilpotent. . Thus ( )= = = . Therefore is also nilpotent. Corollary: A group is nilpotent if and only if is nilpotent. Remark: If and are nilpotent. Then is need not be nilpotent. For example, . , Since is Corollary: Every finite is nilpotent. abelian so it is nilpotent. , } is also abelian, so is also nilpotent. But is not Proof: Every finite is of order for some nilpotent. .The proof is by induction on . If = , then is prime cyclic and so nilpotent. Assume that every group of Proposition Direct product of finitely many nilpotent order , is nilpotent . Let be a group of order groups is nilpotent . Then ≠ . Hence = , . proof: It is sufficient to prove that direct product of two by induction hypothesis is nilpotent and so is nilpotent groups is nilpotent . Suppose and be two nilpotent. groups. Corollary: If All sylow subgroups of a finite group are = normal, then the group is nilpotent. =< │ Proof: If All sylow subgroups are normal, then it is direct ∈ > product of it's sylow subgroups which are prime power =<( ordered groups. Since prime power ordered groups are │ ∈ nilpotent and direct product of nilpotent groups are nilpotent, then the group is nilpotent. < │ (1) Corollary: Maximal normal nilpotent subgroup is the largest nilpotent subgroup. ∈ = = = = . By induction on we have to show that Definition 4: Let be a subgroup of a group . Define a = ∀ . Assume that sequence { ( )| of subgroups of = . =[ , ] inductively as follows: Define ( )= . Assuming that

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA

( ) has already been defined, define ( ) We know that every finite are nilpotent. So = ( ( )). Thus we get a ascending chain all sylow subgroups are nilpotent. Since direct product of = ( )⊴ ( )⊴ ( )⊴ . . . ⊴ ( )⊴ . . . nilpotent groups are nilpotent. Thus is nilpotent. Corollary: A finite nilpotent group is direct product of it's Proposition: Let be a nilpotent group of order . Then sylow subgroups. ( )= for all subgroups of . Theorem 2: ( ). A finite group is nilpotent if Proof: We show by induction on that ( )⊇ and only if all it's maximal subgroups are normal. ∀ . = ⊆ = ( ), = ⊆ ( ) (∵ ⟹ = ). Assume that ( )⊇ . Proof: Let be a finite nilpotent group and a maximal Then to show that ⊆ ( ), or ⇒ subgroup of . Then, since it satisfies normalizer condition, properly contains . Since is ( )= ( ) , or ∈ ( ) ∀ ∈ ( ). maximal, and so is normal. Conversely = . ∈ ⇒ = suppose that all maximal subgroups of are normal. We ⇒ = ∀ ∈ ⇒ show that all sylow subgroups are normal. Let be a = ∀ ∈ ⇒ ∈ ∀ ∈ ⇒ of . Suppose that . ∈ ∀ ∈ ( ) ( ( )⊆ )⇒ ∈ ( ) ( Since is finite there is a maximal subgroup containing ∵ ⊆ ( ))⇒ ∈ ( )∀ ∈ ( )⇒ ∈ . But then, a contradiction to the ( ( )) ∀ ∈ ( )⇒ ∈ ( )∀ . Thus supposition that all maximal subgroups are normal. Hence ⊆ ( ). Since is nilpotent group of index . all sylow subgroups are normal and so is nilpotent. Therefore = . Since ⊆ ( ). ⊆ E. Frattini Subgroup ( )⊆ . Thus = ( )∀ subgroup of . Definition 1: Let be a group. If has no maximal subgroup, then we define the Frattini subgroup of Definition 5: A group is said to satisfy Normalizer denoted by to be itself. If has maximal subgroup Condition if every proper subgroup is properly contained in its normalizer. then Frattini subgroup of is defined to be the intersection of all maximal subgroup of . Thus if Corollary: Every nilpotent group satisfies normalizer denotes the family of all maximal subgroups of , then condition. .

Proof: Let be a nilpotent group of class . Then Example 1: for and are maximal ( )= for all subgroups . If is a proper subgroup subgroups of . for and are and it is not properly contained in ) then ) = maximal subgroups and their intersection is trivial. Also for the only nontrivial proper and so ( )= , a contradiction. normal subgroup of is which is not largest . Proposition: If a finite group satisfies normalizer Definition 2: Let be a group and . Then is said condition, then all its sylow subgroups are normal. to be a non-generator if whenever is generated by and Proof: Suppose that satisfies normalizer condition. Let a set , then = . be a sylow subgroups of . Consider the Theorem 1: The Frattini subgroup is the set of all normalizer of . Then ( ( ))= . Hence non-generators of . ( )= . Since the group satisfies the normalizer condition, cannot be a proper subgroup proof: Suppose is a non-generators of and is a of . Thus and so is normal in . maximal subgroup. If , then is a proper subgroup of , , and so by maximality , = . But is Corollary: A finite group is nilpotent if and only if all a non-generator, and hence which contradicts the it's sylow subgroups are normal in . maximality of (maximal subgroups are always proper). Therefore and as this holds for all maximal Proof: Suppose is nilpotent then all it is Sylow subgroups . Thus . Conversely, suppose subgroups are normal. Thus is direct product of it is , so belongs to all maximal subgroups of ; we sylow subgroups. Conversely suppose all sylow subgroups show that is non-generator. Assume that, for some set are normal. So is direct product of it is Sylow subgroups. , and generates . If , then

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 A Study of Extra Special P-Group there exists a maximal subgroup of which contains II. BASIC FROM LINEAR ALGEBRA , possibly itself, that is, . By Definition 1: A ring ( , +, ) is a nonempty set containing supposition, and so , = , hence at least two elements with two binary operations satisfies which is impossible. Hence and is a non- the following conditions: generator. 1. ( , +) is commutative group whose identity element is denoted by . Proposition: . 2. ( , ) is a semigroup group. 3. The binary operation distributes over + from left as Proof: Let = . = well as from right. = (conjugate of maximal subgroup is maximal)= . Hence . Definition 2: A field ( ,+, ) is a nonempty set at least two elements with two binary operations satisfies the following Proposition: Let is a finite group, is a subgroup of conditions: and = then = . 1. ( ,+) is commutative group whose identity element is denoted by . Proof: If , then there exists a maximal subgroup 2. ( , ) is a commutative group whose identity of which contains poosibly itself. Now element is denoted by . by definition, therefore = , which is 3. The binary operation distributes over + from left as impossible. The result follows. well as from right.

Proposition: (Frattini Argument) . If is a finite group, Definition 3: Let be a ring with identity . A left and is a sylow subgroups of , then module is an abelian group ( , +) together with a map . from to (the image of ( , ) under is denoted by ) such that. Proof: For , = , as . Hence both and are sylow subgroups of (they 1. ( have the same order, and so by they are conjugate 2. in ). Therefore, we can find to satisfy 3. ( ) ) = . Thus , and 4. for all , and , , . so . The result follows because this argument applies to all . In the similar manner we can define right modules. We say that is a left(right) module or is a left(right) Theorem 2: Frattini subgroup of a finite group is nilpotent. module over .

Proof: Let be a finite group and a sylow Remark: If is a commutative ring, then every left of the Frattini subgroup. It is sufficient to module can also be considered as a right module. show that is normal in . If there is In this case we simply say that a module. nothing to do. Suppose that . Using the Frattini Argument we have , as . By Definition 4: A module over a field is called a vector with , we obtain , which gives space over . . The result follows as this holds for all sylow subgroups of . Example 1: Every abelian group ( ,+) is a module.

Theorem 3: A finite group is nilpotent if Example 2: Let be a ring with identity. Then ( ,+) is an and only if . module.

Proof: Let be a finite nilpotent and a maximal Definition 5: Let be a left module. A subset of subgroup. Then from of Wielandt is normal subgroup of is called a submodule of if: Since is maximal, is a group without proper subgroup and so it is prime cyclic, in particular abelian. 1. is a subgroup of ( ,+) But then . This show that . Conversely 2. The map from to induces a map from suppose that . Then every maximal subgroup of to . In other words and contains and so every maximal subgroup is normal. . Thus a submodule is a module at it's own By the of Wielandt is nilpotent. right. In case of vector space submodule are called Subspace.

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA

Definition 6: A left module is said to be finitely 1. , ) = + generated if it has a finite set of generators. A finitely 2. generated vector space over a field is also said to be 3. finite dimensional. For all , , and for all . Thus, if is Definition 7: Let be a nonempty subset of a left bilinear form on , then is a scalar in for each module . An element is called a linear pair ( . combination of members of if: Proposition: Let be a linear form on a vector space . . . + for some , , . . . over a field . , and , , . . . , . 1. For a fixed , the map : given by , ), for all , is linear. Definition 8: A subset of a left module is called 2. For a fixed , the map : given by independent if: , ), for all , is linear. 1. 3. For all , , ) = , ) = 0. 2. Given a finite subset { , , . . . , }of with 4. If , ) = , ) for , , then is a bilinear , for , . . . + form on . . 5. For all , , , , , ) = , )+ A subset which is not independent is called dependent. , ) + , )+ , ).

Definition 9: A subset of module is called Linearly Example 3: In any vector space over a field , there is a Independent if given a finite subset { , , . . . , } of trivial bilinear form given by , )= , where is , , for , . . . + the zero of the field . We will refer to this form as the zero bilinear form on which is identically zero. . A subset which is not linearly independent is called linearly dependent. Example 4: If , the vector space of column vectors or matrices over , then we can get a bilinear form Definition 10: Let be a vector space over a field . A on just like the dot product on by , )= for subset of is called a basis of if it satisfies the following conditions: , . The matrix product is a scalar in .

1. is a maximal linearly independent. Definition 13: Let and be the linear maps defined as: 2. is linearly independent and . , . Then: 3. Every element of is a linear combination of elements of and the representation of an element { │ , ) = for all } as a linear combination of elements of is unique in the sense that if { │ , ) = for all }.

. . . + . . . Let be a bilinear form on a vector space . If + (2) , then call radical of . where , , . . . , are distinct members of , , , . . Definition 14: A bilinear form on a vector space is . , are also distinct members of and , are all said to be non-degenerate if or equivalently nonzero, then . 1. 2. After some rearrangement = Definition 15: A bilinear form on a vector space is 3. S is a minimal set of generators. symmetric if , ) = , ) for all , .

Definition 11: Let be a finitely generated vector space Example 5: In the (example 3.5.21) and the (example over a field . Then the number of elements in a basis of 3.5.22), the bilinear form is symmetric bilinear form. is called Dimension of over . If is not finitely generated then we say that it is infinite dimensional. Example 6: For the vector space , any fixed matrix , a bilinear form on is given by ( , ) = Definition 12: Let be a vector space over a field . A for , .A matrix is said to be map is a bilinear form on if satisfies the symmetric if . Now ( , ). following conditions: On the other hand being a scalar,

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 A Study of Extra Special P-Group

( , ). It follows that ( , ) = Using matrix multiplication, we have , , ( , ) for all for all , . Thus, the symmetric , and , , , matrix forces the bilinear form on to be a . Thus it is non-abelian group. Moreover is symmetric form. the identity of and commutes with all elements of . Also, , have order so any two of them generate the Definition 16: A bilinear form on a vector space is group. Therefore the presentation of is: skew-symmetric if , ) = , ) for all , . , | , . (4) Theorem 1: Let be a finite dimensional vector space over a field . Let be a non-degenerate Putting , , .Thus ( , .) is a group of skew-symmetric bilinear form on . Then order called the Quaternion Group of degree or for some and there exists a basis ={ , , . . . Hamiltonian Group (after the name of Hamilton). { , , , , , . . . } of such that ( , ) = . },{ , , , },{ , , , } and { , , , } are . . , and ( , ) = ( , ) , ( , ) = non trivial proper subgroups of . { , }, 0 . { , }: is a group of order , therefore it is abelian. Thus . Further since is III. EXTRA SPECIAL -GROUP nonabelian, . Hence { , }. . Definition: A finite non-abelian group is called extra Therefore . Thus is extra special special -group if . group.

A. Extra Special -groups of order 8. C. Dihedral group: The dihedral group is one of Here we have discussed extra special -group of order the two non abelian groups of order . An example of is . We have find that there are only two non-isomorphic the symmetry group of the square. Rotating the square extra special -group of order , one is and second is gives rise to four symmetries. We will label them as . follows: Theorem 1: If is a nonabelian group of order = 8 then it is isomorphic to either or .

Proof: Let be a nonabelian group of. The non identity element in have order or . If all elements are of order , then =1 or , = . Thus the group is abelian. Hence there must be an element of order . Suppose and the subgroup generated by is of order . Since has index in , then it is a normal subgroup. Therefore { , , , }. If then has order , so that cannot be. The same thing goes for since that implies that =1, which would mean that has order so that cannot be either. If , then the group is abelian so this is also an impossibility. Therefore . The group has order , so { , , , }. Since has order or , has order or . Thus or . Putting this together, where either , , or , , . In the first case and in the second case .

B. Hamiltonian Group: One of the most famous finite groups is the quaternion group . This group generated by the matrices

Fig 1: Symmetrics of rotation of Dihedral group , , (3)

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA

R0 =Rotation of 0° (no change in position). Proof: Let be nonabelian group of order , is prime. =Rotation of 90° (counterclockwise). Since the center is a subgroup of . Therefore it has orders =Rotation of 180 o. , , or . For , the center never be trivial, =Rotation of 270 o. so it is cannot be of order , thus . Since is H=Rotation of 180 oabout a horizontal axis. not abelian group, therefore . If = , then o V=Rotation of 180 about a vertical axis. = . Thus is cyclic. But by is abelian. o D=Rotation of 180 about the main diagonal. Contradiction with is not abelian group. Thus = D'=Rotation of 180° about the other diagonal. and = .

The improper group of is . Theorem 2: If is nonabelian group of order , then the The prober subgroups of is , commutator subgroup, , = ( , ) | , , , and . has order . . Since HD DH, thus is nonabelian group. The presentation of is: Proof: is normal subgroup of . In a group of order , = . Thus is abelian. Thus , | , . (5) . Since is nonabelian, then is nontrivial. Thus must be proper subgroup of . Therefore the only Also is the identity of and commutes with possibility is . Since the Frattini subgroup all elements of . The operation table for is: contained in the center of a group. Hence .

TABLE 1: OPERATION TABLE FOR D 4 Corollary: Every group of order is extra special - group.

A. Extra Special -group of order A group is said to be the Central product of it's normal subgroups , , . . . , if :

1. . . . . 2. and is direct product of , , . . . , . 3. [ , ] = .

Theorem 1: A finite abelian which every element (non-identity) having order p, which is defined as Further, { , }. { , }: elementary abelian group, can be treated as a vector space is a group of order , therefore it is abelian. Thus over a field . . Further since is nonabelian, . Hence . . Proof: Let be elementary abelian . Define a Therefore . Thus is extra special map from to such that: . group. ( = = + IV. GROUPS OF ORDER For any prime , there are five groups of order up to isomorphism . From the fundamental theorem of abelian = = (6) groups there are three different abelian non-isomorphic groups of order , namely , and = = . These are non isomorphic since they have + = = (7) different maximal orders for their elements. The two nonabelian groups of order have different descriptions for and So will treat these cases separately. ( = = = =

Theorem 1: If is nonabelian group of order , then =. = (8) has order .

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 A Study of Extra Special P-Group

= = (9) VI. REFERENCES [1] A. Mahalanobis, the Diffie-Hellman Key Exchange Theorem 2: Let be a finite extra special group. Then it Protocol and Non-Abelian Nilpotent Groups, Israel J. is central product of non-abelian groups of order . In Math., 165 (2008), 161 - 187. particular is of order for some . [2] A.M. Tripathi, N. Mathu and S. Srivastav, A Study Of Proof: Since . Let , and . Then Nilpotent Groups Through Right Transversals Iranian , )= = = Journal Of Mathematical Sciences And Informatics Vol. 4, . Since ( , ), ( , ) No. 2 (2009), Pp. 49-54. . Thus ( , ) ( , ) =( , ) ( , ). If [3] Adrien Deloro, P-Rank and P-Groups in Algebraic = = , then we have ( , )=( , )=( , ) Groups, Turk J Math 36 (2012), 578 – 582. ( , )= . Thus by induction, we have ( , )= . In particular ( , )= [4] Ahmad M. Alghamdi, Dade’s Projective Conjecture = . Hence = . Thus For P-Block With An Extra-Special Defect Group, is an elementary abelian and so it is a vector International Journal Of Algebra, Vol. 4, 2010, No. 11, 525 space over the field . Since ( , ) = and –534. | |= . Therefore is cyclic group. Let be a generator of such that = ={ , , , . . . [5] Ahmad M. Alghamdi, Ordinary Weight Conjecture , }. Let , and , . Then = , = Implies Brauer’s K(B)-Conjecture, International Journal Of for some , . Now ( , )= = Algebra, Vol. 4, 2010, No. 13, 615 – 618. = =( , ). Thus we have a map [6] Alexander Moret´O, an Answer To Two Questions Of defined by Brewster And Yeh On M-Groups, 2009, Mathematics =( , )= for some . Also we have a map Department University Of Wisconsin 480 Lincoln Drive defined by = . Let : . So : Madison WI 53706 USA. which satisfies = = ( , ). Since is an elementary abelian [7] Andreas Distler, Finite Nilpotent Semigroups Of Small so is bilinear form on a vector space . Since Coclass, Arxiv: 1205.2817v1 [Math.RA] 12 May 2012. ( , )= , so if ( , )= , then ( , )= . Then [8] Artur Schäfer, Masters Thesis, Two Sided and Abelian = . So is skew Group Ring Codes, October 2012, Aachen, Germany. symmetric bilinear form. Next assume that such that = . Then = [9] C. Hopkins, Non-Abelian Groups Whose Groups Of ( , )= = . But ( , )= implies that Isomorphisms Are Abelian, The Annals Of Mathematics, . = the identity element of . 2nd Ser., 29, No. 1/4. (1927 - 1928), 508-520. Hence is non-degenerate. Then for some and there is a basis ={ , , . . . [10] C. J. E. Pinnock, Supersolubility And Some , , , , . . . } such that Characterizations Of Finite Supersoluble Groups, 2nd ( , )= , ( , )= . Edition, January 1998. Also ( , )= = ( , ) . Let [11] Charles Hobby, The Frattini Subgroup Of A P-Group, = . Then each ( ) is a non-abelian 2001. group of order .Since | |= and | | = , so = . So = , , . . . , , , , . . . . Thus [12] Cody Clifton, Commutativity In Non-Abelian Groups, = . . . . which is central product. May 6, 2010.

V. CONCLUSION [13] Craig Gentry, A FULLY HOMOMORPHIC We have solved extra Special -group of order , ENCRYPTION SCHEME, September 2009. where is even prime discussed, we have find that one is [14] D. Jonah, M. Konvisser, Some Non-Abelian P-Groups and second is . We have proved that every non- With Abelian Automorphism Groups, Arch. Math. (Basel) abelian group of order is extra special -group. We 26 (1975), 131 - 133. have proved that be a finite extra special group, then it is central product of non-abelian groups of order . In [15] David A. Craven, the Theory Of P-Groups, Hilary Term, 2008. particular is of order for some .

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA

[16] David Bornand, Elementary Abelian Subgroups In P- [32] M. Morigi, On The Minimal Number Of Generators Groups Of Class 2, 23 July 2009. Of Finite Non-Abelian P-Groups Having An Abelian Automor- Phism Group, Comm. Algebra 23 (1995), 2045 - [17] David John Green and Pham Anh Minh, Almost All 2065. Extraspecial P-Groups Are Swan Groups, Ath/9911083v1 [Math.AT] 12 Nov 1999. [33] Mohammad K. Azarian, Conjectures And Questions Regarding Near Frattini Subgroups Of Generalized Free [18] Doron Hai-Reuven, Non-Solvable Graph Of A Finite Products Of Groups, International Journal Of Algebra, Vol. Group And Solvabilizers, Rxiv: 1307.2924v1 [Math.GR] 5, 2011, No. 1, 1 – 15. 10 Jul 2013. [34] P. Hegarty, Minimal Abelian Automorphism Groups [19] G. A. Miller, A Non-Abelian Group Whose Group Of Of Finite Groups, Rend. Sem. Mat. Univ. Padova 94 Isomorphism Is Abelian, Mess. Of Math. 43 (1913-1914), (1995), 121 - 135. 124 - 125. [35] Pham Anh Minh, the Modp Cohomology Group Of [20] Geir T. Helleloid And Ursula Martin, The Extra-Special P-Group Of Order P5 And Of Exponent P2, Automorphism Group Of A Finite P-Group Is Almost Math. Proc. Camb. Phil. Soc. (1996), Printed In Great Always A P-Group, Nankai University, Tianjin, China, Britain. 2007. [36] Ryan J. Wisnesky, Solvable Groups, May 2005. [21] Geir T. Helleloid, Automorphism Groups Of Finite P- Groups: Structure and Applications, Arxiv: 0711.2816v1 [37] Stuart Hendren, Extra Special Defect Groups, [Math.GR] 18 Nov 2007. September 2003, A Thesis Submitted To The University Of Birmingham For The Degree Of Doctor Of Philosophy. [22] H. Heineken, H. Liebeck, The Occurrence Of Finite Groups In The Automorphism Group Of Nilpotent Groups [38] V. K. Jain, M. K. Yadav, On Finite P-Groups Who’s Of Class 2, Arch. Math. (Basel) 25 (1974), 8 - 16. Automorphisms Are All Central, Israel J. Math. 189 (1979/80), 497 - 503. (2012),225 - 236.

[23] H. Hilton, an Introduction To The Theory Of Groups [39] Wojciech A. Trybulec, Lattice Of Subgroups Of A Of Finite Order, Oxford, Clarendon Press (1908). Group. Frattini Subgroup, Formalized Mathematics Vol.2, No.1, January–February 1991. [24] I. M. Isaacs, the Fitting and Frattini Subgroups, 2002. [40] Y. Berkovich, Z. Janko, Groups Of Prime Power [25] J. E. Adney, T. Yen, Automorphisms Of A P-Group, Order - Volume 2, Walter De Gruyter, Berlin, New York Illinois J. Math. 9 (1965), 137 - 143. (2008).

[26] James Clark Beidleman And Tae Kun Seo, [41] Yark, Frattini Subgroup Of A Finite Group Is Generalized Frattini Subgroups Of Finite Groups, Pacific Nilpotent, 2013-03-21. Journal Of Mathematics, Vol. 23, No. 3, 1967.

[27] Javier Otal, the Frattini Subgroup Of A Group, Spain, 2001.

[28] John C. Lennox, Finite Frattini Factors In Finitely Generated Soluble Groups1, Proceedings Of The American Mathematical Society Volume 41, Number 2, December 1973.

[29] John Cossey and Alice Whittemore1, On the Frattini Subgroup, 2001.

[30] Katrin E. Gehles, Ordinary Characters Of Finite Special Linear Groups, M.Sc. Dissertation August 2002.

[31] M. H. Jafari, Elementary Abelian P-Groups as Central Automorphism Groups, Comm. Algebra 34 (2006), 601 - 607.

International Journal of Scientific Engineering and Technology Research Volume.02, IssueNo.19, December-2013, Pages:2223-2234