On the Chermak-Delgado Lattices of Split Metacyclic p-Groups

On the Chermak-Delgado Lattices of Split Metacyclic p-Groups Research by Brianne Power, Erin Brush, and Kendra Johnson-Tesch Supervised by Jill Dietz at St. Olaf College

Background Chermak and Delgado (1989) were interested in finding families of characteristic subgroups. They introduced a measure that was later deemed the “Chermak-Delgado” measure. The subgroups with maximal Chermak-Delgado measure form a lattice. Not many Chermak-Delgado lattices have been unearthed due to their complexity. These lattices give a visual representation of deep structural properties of finite p-groups and their subgroups. Andrew Chermak Kansas State University Alberto Delgado Illinois State University

Useful Definitions Center of G:The set of elements in a group G that commute with every element in G Z(G) = { z ϵ G | zg = gz for all g ϵ G } Centralizer of S: The set of elements in G that commute with all of the elements in a subset S of G CG(S) = { c ϵ G | sc = cs for all s ϵ S }

Subgroup Lattice of G G H1 H3 H2 H4 < H3 H4 H5 e

The Chermak-Delgado Measure The Chermak-Delgado measure of a subgroup H is G is mG(H) = |H| |CG(H)|. We write m*(G) to denote the largest possible Chermak-Delgado measure of the subgroups of G.

The Chermak-Delgado Lattice The Chermak-Delgado lattice of a finite group G is a lattice comprised of subgroups of G with the largest possible Chermak-Delgado measure. For the finite group G, we write CD(G) for the Chermak-Delgado lattice of G.

Example 1: The abelian group Z6 G = Z6 = {0, 1, 2, 3, 4, 5} H1 = {0, 2, 4} H2 = {0, 3} H3 = {0} mG(G) = |G| |CG(G)| = |G| |Z(G)| = |G|2 = 62 = 36 ← m*(G) mG(H1) = |H1| |CG(H1)| = |H1| |G| = 3.6 = 18 mG(H2) = |H2| |CG(H2)| = |H2| |G| = 2.6 = 12 mG(H3) = |H3| |CG(H3)| = |H3| |G| = 1.6 = 6

Generalization: Abelian Groups Let A be an abelian group. m*(A) = mA(A) = |A| |CA(A)| = |A| |Z(A)| = |A|2

(the dihedral group of order 8) Example 2: Dihedral group D8 Presentation of D8:< x, y | x4 = 1 = y2, yxy-1 = x3 > Rotation Reflection x y

mG(G) = |G| |Z(G)| = |G| |H6| = 8.2 = 16 mG(H1 ) = 16 mG(H2 ) = 16 mG(H3 ) = 16 mG(H4 ) = 8 mG(H5 ) = 8 mG(H6 ) = 16 mG(H7 ) = 8 mG(H8 ) = 8 mG(e) = 8 Example 2: Dihedral group D8 G = D8 H1 = <x2,y> H2 = <x> H3 = <x2,xy> H4 = <y> H5 = <x2y> H6 = <x2> H7 = <xy> H8 = <x3y> e m*(G)=16

Subgroup Lattice D8 Chermak-Delgado Lattice <x2,y> <x2,xy> <x> <xy> <x2> <y> <x3y> <x2y> e

Example 3: Dihedral group D12 mG(G) = 24 CD(D12): <r> mG(<r>) = 36 = m*(G)

Metacyclic p-Groups • G is metacyclic if it has a cyclic normal subgroup N such that G/N is also cyclic • Metacylic groups are generated by two elements x and y where: • x generates N • yN generates G/N • A metacylic p-group has pk elements (p a prime)

CD lattice of P(p,m)

P(p,m): How to Prove • Gather information about all subgroups of P • Find centralizers using known relations • Apply properties of p-groups and normal subgroups

Generalize to other metacyclic groups P(p,m) = < x, y | xp^m = 1 = yp, yxy-1 = x1+p^(m-1) > P(p,m,1,1) =< x, y | xp^m = 1 = yp^1, yxy-1 = x1+p^(m-1) > P(p,m,n,r) =< x, y | xp^m = 1 = yp^n, yxy-1 = x1+p^(m-r) >

A Broader Family of Metacyclics P(p,m,n,r) =< x, y | xp^m = 1 = yp^n, yxy-1 = x1+p^(m-r) > where m > 2, n > 1, and 1 < r < min{m-1, n} Observations: |P| = pm+n and Z(P) = <xp^r, yp^r> Theorem: mP(P) = p2(m+n-r) mP(P) ≟m*(P)

The sublattice Note: Hab = < xp^a, yp^b >

A Broader Family of Metacyclics Theorem: m*(P) = p2(m+n-r) = mP(P) This means that the lattice is a sublattice of CD(P)!

P(p,m,n,r): How we found the lattice • Used examples and tested out patterns • Applied properties of p-groups and normal subgroups • External research confirmed that the measure of these groups is the maximal measure of P

Current Research • Confirmation that our lattice is a sublattice of CD(P) • What else is in CD(P)? • What does the lattice of all subgroups of P look like? • Investigate other measures identified by Chermak and Delgado

Research Sources • L. An, J. Brennan, H. Qu, and E. Wilcox, Chermak-Delgado lattice extension theorems, submitted, 2013. http://arxiv.org/pdf/1307.0353v1.pdf • Y. Berkovich, Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic, Israel J. Math. 194 (2013), 831-869. • J.N.S. Bidwell and M.J. Curran, The automorphism group of a split metacyclic p-group, Math. Proc. R. Ir. Acad. 110A (2010), no. 1, 57-71. • B. Brewster, P. Hauck, and E. Wilcox, Groups whose Chermak-Delgado lattice is a chain, submitted, 2013. http://arxiv.org/pdf/1305.2327v1.pdf • B. Brewster and E. Wilcox, Some groups with computable Chermak-Delgado lattices, Bull. Aus. Math. Soc. {86 (2012), 29-40. • A. Chermak and A. Delgado, A measuring argument for finite groups, Proc. AMS 107 (1989), no. 4, 907-914. • G. Glauberman, Centrally large subgroups of finite p-groups, J. Algebra 300 (2006), no. 2, 480-508. • L. Héthelyi and B, Külshammer, Characters, conjugacy classes and centrally large subgroups of p-groups of small rank, J. Algebra 340 (2011), 199-210. • I. M. Isaacs, Finite Group Theory, American Mathematical Society, 2008. • King, Presentations of Metacyclic Groups, Bull. Aus. Math. Soc. 8 (1973), 103-131. • W.K. Nicholson, Introduction to Abstract Algebra, 4th Edition, Wiley, 2012. • M. Schulte, Automorphisms of metacyclic p-groups with cyclic maximal subgroups, Rose-Hulman Undergraduate Research Journal 2 (2001), no. 2. • M. Suzuki, Group Theory II, Springer-Verlag, 1986.

Image Sources http://www.math.ksu.edu/people/personnel_detail?person_id=1326 https://faculty.sharepoint.illinoisstate.edu/aldelg2/Pages/default.aspx http://www.quickmeme.com/Bad-Joke-Eel/page/565/ http://fergalsresearch.weebly.com/subgroup-lattices.html

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On the Chermak-Delgado Lattices of Split Metacyclic p-Groups