Computing Chief Series, Composition Series and Socles in Large Permutation Groups
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THE UPPER BOUND of COMPOSITION SERIES 1. Introduction. a Composition Series, That Is a Series of Subgroups Each Normal in the Pr
THE UPPER BOUND OF COMPOSITION SERIES ABHIJIT BHATTACHARJEE Abstract. In this paper we prove that among all finite groups of or- α1 α2 αr der n2 N with n ≥ 2 where n = p1 p2 :::pr , the abelian group with elementary abelian sylow subgroups, has the highest number of composi- j Pr α ! Qr Qαi pi −1 ( i=1 i) tion series and it has i=1 j=1 Qr distinct composition pi−1 i=1 αi! series. We also prove that among all finite groups of order ≤ n, n 2 N α with n ≥ 4, the elementary abelian group of order 2 where α = [log2 n] has the highest number of composition series. 1. Introduction. A composition series, that is a series of subgroups each normal in the previous such that corresponding factor groups are simple. Any finite group has a composition series. The famous Jordan-Holder theo- rem proves that, the composition factors in a composition series are unique up to isomorphism. Sometimes a group of small order has a huge number of distinct composition series. For example an elementary abelian group of order 64 has 615195 distinct composition series. In [6] there is an algo- rithm in GAP to find the distinct composition series of any group of finite order.The aim of this paper is to provide an upper bound of the number of distinct composition series of any group of finite order. The approach of this is combinatorial and the method is elementary. All the groups considered in this paper are of finite order. 2. Some Basic Results On Composition Series. -
Normal Subgroup Structure of Totally Disconnected Locally Compact Groups
Normal subgroup structure of totally disconnected locally compact groups Colin D. Reid Abstract The present article is a summary of joint work of the author and Phillip Wesolek on the normal subgroup structure of totally disconnected locally compact second-countable (t.d.l.c.s.c.) groups. The general strategy is as follows: We obtain normal series for a t.d.l.c.s.c. group in which each factor is ‘small’ or a non-abelian chief factor; we show that up to a certain equivalence relation (called association), a given non-abelian chief factor can be inserted into any finite normal series; and we obtain restrictions on the structure of chief factors, such that the restrictions are invariant under association. Some limitations of this strategy and ideas for future work are also discussed. 1 Introduction A common theme throughout group theory is the reduction of problems concern- ing a group G to those concerning the normal subgroup N and the quotient G=N, where both N and G=N have some better-understood structure; more generally, one can consider a decomposition of G via normal series. This approach has been espe- cially successful for the following classes of groups: finite groups, profinite groups, algebraic groups, connected Lie groups and connected locally compact groups. To summarise the situation for these classes, let us recall the notion of chief factors and chief series. Definition 1. Let G be a Hausdorff topological group. A chief factor K=L of G is a pair of closed normal subgroups L < K such that there are no closed normal subgroups of G lying strictly between K and L.A descending chief series for G is a Colin D. -
REPRESENTATIONS of ELEMENTARY ABELIAN P-GROUPS and BUNDLES on GRASSMANNIANS
REPRESENTATIONS OF ELEMENTARY ABELIAN p-GROUPS AND BUNDLES ON GRASSMANNIANS JON F. CARLSON∗, ERIC M. FRIEDLANDER∗∗ AND JULIA PEVTSOVA∗∗∗ Abstract. We initiate the study of representations of elementary abelian p- groups via restrictions to truncated polynomial subalgebras of the group alge- p p bra generated by r nilpotent elements, k[t1; : : : ; tr]=(t1; : : : ; tr ). We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several gen- eral constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples. Contents 1. The r-rank variety Grass(r; V)M 4 2. Radicals and Socles 10 3. Modules of constant radical and socle rank 15 4. Modules from quantum complete intersections 20 5. Radicals of Lζ -modules 27 6. Construction of Bundles on Grass(r; V) 35 6.1. A local construction of bundles 36 6.2. A construction by equivariant descent 38 7. Bundles for GLn-equivariant modules. 43 8. A construction using the Pl¨ucker embedding 51 9. APPENDIX (by J. Carlson). Computing nonminimal 2-socle support varieties using MAGMA 57 References 59 Quillen's fundamental ideas on applying geometry to the study of group coho- mology in positive characteristic [Quillen71] opened the door to many exciting de- velopments in both cohomology and modular representation theory. Cyclic shifted subgroups, the prototypes of the rank r shifted subgroups studied in this paper, were introduced by Dade in [Dade78] and quickly became the subject of an intense study. -
Topics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. -
On Generalizations of Sylow Tower Groups
Pacific Journal of Mathematics ON GENERALIZATIONS OF SYLOW TOWER GROUPS ABI (ABIADBOLLAH)FATTAHI Vol. 45, No. 2 October 1973 PACIFIC JOURNAL OF MATHEMATICS Vol. 45, No. 2, 1973 ON GENERALIZATIONS OF SYLOW TOWER GROUPS ABIABDOLLAH FATTAHI In this paper two different generalizations of Sylow tower groups are studied. In Chapter I the notion of a fc-tower group is introduced and a bound on the nilpotence length (Fitting height) of an arbitrary finite solvable group is found. In the same chapter a different proof to a theorem of Baer is given; and the list of all minimal-not-Sylow tower groups is obtained. Further results are obtained on a different generalization of Sylow tower groups, called Generalized Sylow Tower Groups (GSTG) by J. Derr. It is shown that the class of all GSTG's of a fixed complexion form a saturated formation, and a structure theorem for all such groups is given. NOTATIONS The following notations will be used throughont this paper: N<]G N is a normal subgroup of G ΛΓCharG N is a characteristic subgroup of G ΛΓ OG N is a minimal normal subgroup of G M< G M is a proper subgroup of G M<- G M is a maximal subgroup of G Z{G) the center of G #>-part of the order of G, p a prime set of all prime divisors of \G\ Φ(G) the Frattini subgroup of G — the intersec- tion of all maximal subgroups of G [H]K semi-direct product of H by K F(G) the Fitting subgroup of G — the maximal normal nilpotent subgroup of G C(H) = CG(H) the centralizer of H in G N(H) = NG(H) the normalizer of H in G PeSy\p(G) P is a Sylow ^-subgroup of G P is a Sy-subgroup of G PeSγlp(G) Core(H) = GoreG(H) the largest normal subgroup of G contained in H= ΓioeoH* KG) the nilpotence length (Fitting height) of G h(G) p-length of G d(G) minimal number of generators of G c(P) nilpotence class of the p-group P some nonnegative power of prime p OP(G) largest normal p-subgroup of G 453 454 A. -
Let G Be a Group. a Subnormal Series of Subgroups of G Is a Sequence G {E} < G 1 < ... < Gn = G Such That G I−1
NORMAL AND SUBNORMAL SERIES Let G be a group. A subnormal series of subgroups of G is a sequence G0 = feg < G1 < ::: < Gn = G such that Gi−1 / Gi for i = 1; :::; n. A normal series is a subnormal series such that each Gi is normal in G. The number n is called the length of the (sub)normal series. A (sub)normal series is called proper if all the quotient groups Gi=Gi−1 are nontrivial. These quotient groups are refered to as successive quotients of the (sub)normal series. We say that two (sub)normal series are equivalent if they have the same length and there is a permutation π 2 Sn such that the groups Gi=Gi−1 and Gπ(i)=Gπ(i)−1 are isomorphic for all i = 1; 2; :::; n. A refinement of a (sub)normal series G0 = feg < G1 < ::: < Gn = G is a 0 0 0 (sub)normal series G0 = feg < G1 < ::: < Gm = G such that there is a strictly 0 increasing function f : f1; :::; ng −! f1; 2; :::; mg such that Gi = Gf(i) for all i. We prove now the following important theorem: Schreier Refinement Theorem. Any two (sub)normal series in G have equivalent refinements. It the series are proper, the refinements can be chosen proper as well. Proof: Note first that each integer k can be uniquely written in the form (m+1)i+j for some integer i and some j 2 f0; :::; mg. If 0 ≤ k < (m+1)(n+1) then 0 ≤ i ≤ n. Similarly, each integer 0 ≤ k < (m + 1)(n + 1) can be uniquely written in the form (n + 1)j + i, where 0 ≤ i ≤ n and 0 ≤ j ≤ m. -
Composition Series of Arbitrary Cardinality in Abelian Categories
COMPOSITION SERIES OF ARBITRARY CARDINALITY IN ABELIAN CATEGORIES ERIC J. HANSON AND JOB D. ROCK Abstract. We extend the notion of a composition series in an abelian category to allow the mul- tiset of composition factors to have arbitrary cardinality. We then provide sufficient axioms for the existence of such composition series and the validity of “Jordan–Hölder–Schreier-like” theorems. We give several examples of objects which satisfy these axioms, including pointwise finite-dimensional persistence modules, Prüfer modules, and presheaves. Finally, we show that if an abelian category with a simple object has both arbitrary coproducts and arbitrary products, then it contains an ob- ject which both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. Contents 1. Introduction 1 2. Background 4 3. Subobject Chains 6 4. (Weakly) Jordan–Hölder–Schreier objects 12 5. Examples and Discussion 21 References 28 1. Introduction The Jordan–Hölder Theorem (sometimes called the Jordan–Hölder–Schreier Theorem) remains one of the foundational results in the theory of modules. More generally, abelian length categories (in which the Jordan–Hölder Theorem holds for every object) date back to Gabriel [G73] and remain an important object of study to this day. See e.g. [K14, KV18, LL21]. The importance of the Jordan–Hölder Theorem in the study of groups, modules, and abelian categories has also motivated a large volume work devoted to establishing when a “Jordan–Hölder- like theorem” will hold in different contexts. Some recent examples include exact categories [BHT21, E19+] and semimodular lattices [Ro19, P19+]. In both of these examples, the “composition series” arXiv:2106.01868v1 [math.CT] 3 Jun 2021 in question are assumed to be of finite length, as is the case for the classical Jordan-Hölder Theorem. -
The M\" Obius Number of the Socle of Any Group
Contents 1 Definitions and Statement of Main Result 2 2 Proof of Main Result 3 2.1 Complements ........................... 3 2.2 DirectProducts.......................... 5 2.3 The Homomorphic Images of a Product of Simple Groups . 6 2.4 The M¨obius Number of a Direct Power of an Abelian Simple Group ............................... 7 2.5 The M¨obius Number of a Direct Power of a Nonabelian Simple Group ............................... 8 2.6 ProofofTheorem1........................ 9 arXiv:1002.3503v1 [math.GR] 18 Feb 2010 1 The M¨obius Number of the Socle of any Group Kenneth M Monks Colorado State University February 18, 2010 1 Definitions and Statement of Main Result The incidence algebra of a poset P, written I (P ) , is the set of all real- valued functions on P × P that vanish for ordered pairs (x, y) with x 6≤ y. If P is finite, by appropriately labeling the rows and columns of a matrix with the elements of P , we can see the elements of I (P ) as upper-triangular matrices with zeroes in certain locations. One can prove I (P ) is a subalgebra of the matrix algebra (see for example [6]). Notice a function f ∈ I(P ) is invertible if and only if f (x, x) is nonzero for all x ∈ P , since then we have a corresponding matrix of full rank. A natural function to consider that satisfies this property is the incidence function ζP , the characteristic function of the relation ≤P . Clearly ζP is invertible by the above criterion, since x ≤ x for all x ∈ P . We define the M¨obius function µP to be the multiplicative inverse of ζP in I (P ) . -
A Splitting Theorem for Linear Polycyclic Groups
New York Journal of Mathematics New York J. Math. 15 (2009) 211–217. A splitting theorem for linear polycyclic groups Herbert Abels and Roger C. Alperin Abstract. We prove that an arbitrary polycyclic by finite subgroup of GL(n, Q) is up to conjugation virtually contained in a direct product of a triangular arithmetic group and a finitely generated diagonal group. Contents 1. Introduction 211 2. Restatement and proof 212 References 216 1. Introduction A linear algebraic group defined over a number field K is a subgroup G of GL(n, C),n ∈ N, which is also an affine algebraic set defined by polyno- mials with coefficients in K in the natural coordinates of GL(n, C). For a subring R of C put G(R)=GL(n, R) ∩ G.LetB(n, C)andT (n, C)bethe (Q-defined linear algebraic) subgroups of GL(n, C) of upper triangular or diagonal matrices in GL(n, C) respectively. Recall that a group Γ is called polycyclic if it has a composition series with cyclic factors. Let Q be the field of algebraic numbers in C.Every discrete solvable subgroup of GL(n, C) is polycyclic (see, e.g., [R]). 1.1. Let o denote the ring of integers in the number field K.IfH is a solvable K-defined algebraic group then H(o) is polycyclic (see [S]). Hence every subgroup of a group H(o)×Δ is polycyclic, if Δ is a finitely generated abelian group. Received November 15, 2007. Mathematics Subject Classification. 20H20, 20G20. Key words and phrases. Polycyclic group, arithmetic group, linear group. -
Extreme Classes of Finite Soluble Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 9, 285-313 (1968) Extreme Classes of Finite Soluble Groups ROGER CARTER The Mathematics Institute, University of Warwick, Coventry, England BERND FISCHER Mathematisches Seminar der Universitdt Frankfurt am Main, Germany TREVOR HAWKES Corpus Christi College, Cambridge, England Communicated by B. Huppert Received August 24, 1967 Knowledge of the structure of a specified collection 9 of subgroups of a group G can often yield detailed information about G itself. For example we have the elementary fact that if all the minimal cyclic subgroups of a group G are n-groups then G is a n-group. Another example is provided by an unpublished theorem of M. B. Powell who has shown that if we take for 9’ the collection of 2-generator subgroups of a finite soluble group G then G has p-length at most n if and only if every member of Y has p-length at most n. Clearly one should aim to make the set Y as small as possible con- sistent with still retrieving useful information about the structure of G. In this note we investigate a situation of this type and for the purpose intro- duce a class (3 of extreme groups. (5 is a subclass of the class of finite soluble groups and loosely speaking comprises those groups which contain as few complemented chief factors, as possible, in a given chief series; more precisely, a group G is extreme if it has a chief series with exactly Z(G) complemented chief factors, where Z(G) denotes the nilpotent (or Fitting) length of G. -
Classifying Categories the Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories
U.U.D.M. Project Report 2018:5 Classifying Categories The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories Daniel Ahlsén Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2018 Department of Mathematics Uppsala University Classifying Categories The Jordan-Holder¨ and Krull-Schmidt-Remak theorems for abelian categories Daniel Ahlsen´ Uppsala University June 2018 Abstract The Jordan-Holder¨ and Krull-Schmidt-Remak theorems classify finite groups, either as direct sums of indecomposables or by composition series. This thesis defines abelian categories and extends the aforementioned theorems to this context. 1 Contents 1 Introduction3 2 Preliminaries5 2.1 Basic Category Theory . .5 2.2 Subobjects and Quotients . .9 3 Abelian Categories 13 3.1 Additive Categories . 13 3.2 Abelian Categories . 20 4 Structure Theory of Abelian Categories 32 4.1 Exact Sequences . 32 4.2 The Subobject Lattice . 41 5 Classification Theorems 54 5.1 The Jordan-Holder¨ Theorem . 54 5.2 The Krull-Schmidt-Remak Theorem . 60 2 1 Introduction Category theory was developed by Eilenberg and Mac Lane in the 1942-1945, as a part of their research into algebraic topology. One of their aims was to give an axiomatic account of relationships between collections of mathematical structures. This led to the definition of categories, functors and natural transformations, the concepts that unify all category theory, Categories soon found use in module theory, group theory and many other disciplines. Nowadays, categories are used in most of mathematics, and has even been proposed as an alternative to axiomatic set theory as a foundation of mathematics.[Law66] Due to their general nature, little can be said of an arbitrary category. -
On 2-Closures of Rank 3 Groups
On 2-closures of rank 3 groups S.V. Skresanov∗ Abstract A permutation group G on Ω is called a rank 3 group if it has pre- cisely three orbits in its induced action on Ω×Ω. The largest permutation group on Ω having the same orbits as G on Ω × Ω is called the 2-closure of G. A description of 2-closures of rank 3 groups is given. As a spe- cial case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional. 1 Introduction Let G be a permutation group on a finite set Ω. Recall that the rank of G is the number of orbits in the induced action of G on Ω Ω; these orbits are called 2-orbits. If a rank 3 group has even order, then its non-diagonal× 2-orbit induces a strongly regular graph on Ω, which is called a rank 3 graph. It is readily seen that a rank 3 group acts on the corresponding rank 3 graph as an automorphism group. Notice that an arc-transitive strongly regular graph need not be a rank 3 graph, since its automorphism group might be intransitive on non-arcs. Related to this is the notion of a 2-closure of a permutation group. The group G(2) is the 2-closure of a permutation group G, if G(2) is the largest permutation group having the same 2-orbits as G. Clearly G G(2), the 2-closure of G(2) is again G(2), and G(2) has the same rank as G.