Introduction to D8 × D8 and Its Subgroup Lattice Bria Morgan and Dan Schilcher Faculty Mentor: Dandrielle Lewis University of Wisconsin-Eau Claire

Introduction to D8 × D8 and its Subgroup Lattice Bria Morgan and Dan Schilcher Faculty Mentor: Dandrielle Lewis University of Wisconsin-Eau Claire

1. Purpose and History 2. Properties of D8 3. Goursat’s Theorem 4. Subgroup Containment

Let A and B be groups and consider the direct product A × B. Theorem 1 [3] Theorem 2 [4] What can one say about the subgroups of A × B? In 1889, Edouard D8, the dihedral group of order 8, is an extraspecial group. It has Let A and B be groups. Then there exists a bijection Suppose U2, U 1 ≤ A × B, where U1 is given by the Goursat proved a theorem that provides the structure of subgroups the following presentation: between the set of all subgroups of A × B and the I1 L1 in a direct product. We have applied the subgroup containment triple , , σ 1 and U2 is given by the triple 4 2 I L I J1 K1  theorem given in [4], which uses the same notation as Goursat’s D8 =< r, s | r = s = 1 , rsr = s > set of all triples , , σ , where is a factor group J K J I L Theorem, to the specific case of D ×D . Our intention is to improve   2 2 8 8 L I L , , σ 2 . Then U2 ≤ U1 if and only if: and make more effective this containment theorem for further use. The subgroup lattice of D8 and our notation for its subgroups are of A, K is a factor group of B, and σ : J → K J2 K2  given below. is an isomorphism between the factor groups. 1. I2 ≤ I1, J 2 ≤ J1, L 2 ≤ L1, and K2 ≤ K1 Definition A p-group P is extraspecial if the center of P , the 2 3 2 3 I J L K Frattini subgroup of P , and the commutator subgroup of P are all D8 = hr, s i = {1, r , r , s, rs, r s, r s} 2 1 2 1 1. Let U ≤ A × B and I , L , σ the triple associated with it. 2. σ1 = the same. 2 2 3 J K  J  K G1 = hr , rs i = {1, r , rs, r s}   1 1 G = hr2, s i = {1, r 2, s, r 2s} 2. Consider the projections πA : A × B → A and πB : A × B → B. Extraspecial groups have been classified[1], and there are exactly two 2 D I2 ∩ J1 L2 ∩ K1 8? 3. σ = 2 3  ?? 2  ?? 3. It follows that U ∩ A / π (U), U ∩ B / π (U), and R = hri = {1, r, r , r }  ?? A B  J  K extraspecial groups of order 32. In the subgroup lattice of D8 × D8,  ?? 2 2 2 2  ? πA(U) πB(U) which we are constructing, there is an extraspecial group of order Z = hr i = {1, r } G1 R G2 σ : → is an isomorphism.  ??  ?? 4.  ??  ?? U ∩ A U ∩ B  ??  ?? 32, which is given by the central product of D8 with D8.  ??  ?? I J L K H1 = hrs i = {1, rs }  ??  ?? 2 1 2 1  ?  / 3 3 4. Let I = πA(U), J = U ∩ A, L = πB(U), and K = U ∩ B. H O H Z H H σ1 H = hr si = {1, r s} 1 OOO 2? 3 ooo 4 J K 2 OO ??  oo 1 1 OOO ??  ooo Definition A central product is a quotient of a direct product with OOO ??  ooo 2 2 OOO ??  ooo OOO ??  ooo 5. The subgroup structure given by Goursat’s Theorem is H3 = hr si = {1, r s} OOO? ooo θ e θ an amalgamated center. O 1 oo U = {(a, b ) | a ∈ I, b ∈ L, and σ(aJ ) = bK }. 1 2 H4 = hsi = {1, s }   I2 σ L2 1 = h1i = {1} 2/ I ∩ J L ∩ K 2 1 e 2 1

5. Results 6. Results, continued 8. Lattice

To construct this subgroup lattice, we first identified all factor groups of D8. Secondly, Below is the subgroup lattice that we have constructed so far, including the subgroups of orders 1, 2, 4, and 8 we counted the number of subgroups by identifying all possible isomorphic factor groups Notation Order I,J L,K # of Subgps. Group Structure Max. Subgroups ←− ←− ←− ←− read from bottom to top respectively. and finding the automorphism group of the factor groups. This method is given in [5]. D8 8 D8, D 8 1,1 1 D8 G1, G2, G3 |I| · | L| −→ −→ −→ −→ We calculated the order of each subgroup using |U| = , given in [4]. We D8 8 1,1 D8, D 8 1 D8 G1, G2, G3 |I/J | ←−− ←− ←− concluded that E3,i 8 R,R Hi, H i 5 C4XC 2 V5,i , G3, F3,i D8 × D8 has 389 subgroups: 1 ≤ i ≤ 5 1 subgroup of order 64 −−→ −→ −→ 15 subgroups of order 32 E3,i 8 Hi, H i R,R 5 C2XC 4 Vi, 5, G3, F3,i 67 subgroups of order 16 1 ≤ i ≤ 5 143 subgroups of order 8 127 subgroups of order 4 35 subgroups of order 2 7. Future Research 1 subgroup of order 1 Going forward, our next order of business is to finish determining subgroup containment We developed notation for each subgroup based on the projections and intersections for the groups of order 16 and 32. This will allow us to complete construction of the which define it. This assisted in organizing the lattice and recognizing group structures. subgroup lattice of D8 × D8. Once this is complete, we will identify the extraspecial Given that we are working in a p-group, determining the maximal subgroups of each group of order 32. subgroup is sufficient to give the group’s structure. Thus far, we have determined the group structure and maximal subgroups for all subgroups of orders 1, 2, 4 and 8. A In addition, this process of determining subgroup containment can be applied to any chart with a sampling of the notation, number of subgroups with that notation, group direct product of groups and is highly methodical. We would like to make it more structure, and maximal subgroups of a few groups of order 8 follow in the next panel. effective by possibly writing a program to implement it.

References Acknowledgments

[1] Doerk, K. & Hawkes, T. (1992). Finite Soluble Groups (De Gruyter Expositions in Mathematics). Walter De Gruyter and Co. • Office of Research and Sponsored Programs, UW-Eau Claire [2] Dummit, D.S. & Foote, R.M. (2004). Abstract Algebra, Third Edition. John Wiley and Sons, Inc. • Subgroup lattice drawn in Geometer’s Sketchpad. [3] Goursat, E. (1889). Sur les substitutions orthogonales et les divisions r`eguli´eres de l’espace. Ann. Sci. Ecole` Norm. Sup. 6 p. 9-102 • Images created and rendered in L AT X. [4] Lewis, D. (2011). Containment of Subgroups in a Direct Product of Groups . Doctoral dissertation, Binghamton University. E [5] Petrillo, J. (2011). Counting Subgroups in a Direct Product of Finite Cyclic Groups. College Math J. 42 (3) p. 215-222