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

Brianne Power,Erin Brush, and Kendra Johnson-TeschSupervised by Jill Dietz at St. Olaf

College

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 maximalChermak-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.

Background

Andrew ChermakKansas State University

Alberto DelgadoIllinois 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

e

H2

H3

H5

H4

H4 < H3

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 writeCD(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 mG(H1) = |H1| |CG(H1)| = |H1| |G| = 3.6 = 18mG(H2) = |H2| |CG(H2)| = |H2| |G| = 2.6 = 12mG(H3) = |H3| |CG(H3)| = |H3| |G| = 1.6 = 6

← m*(G)

Generalization: Abelian Groups

Let A be an abelian group.

m*(A) = mA(A)

= |A| |CA(A)|

= |A| |Z(A)|

= |A|2

Example 2: Dihedral group D8

Presentation of D8: < x, y | x4 = 1 = y2, yxy-1 = x3 >

(the dihedral group of order 8)

x y

Rotation

Reflection

G = D8

H1 = <x2,y>H2 = <x>H3 = <x2,xy>H4 = <y>H5 = <x2y>H6 = <x2>H7 = <xy>H8 = <x3y>e

Example 2: Dihedral group D8

mG(G) = |G| |Z(G)| = |G| |H6| = 8.2 = 16

mG(H1 ) = 16mG(H2 ) = 16mG(H3 ) = 16mG(H4 ) = 8mG(H5 ) = 8mG(H6 ) = 16mG(H7 ) = 8mG(H8 ) = 8mG(e) = 8

m*(G)=16

D8

<x2,y> <x> <x2,xy>

<x2> <x3y><y> <x2y> <xy>

e

Subgroup Lattice Chermak-Delgado

Lattice

CD(D12): <r>

Example 3: Dihedral group D12

mG(G) = 24

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:o x generates No yN generates G/N

● A metacylic p-group has pk elements (p a prime)

P(p,m): A family of metacyclic p-groups

P(p,m) = < x, y | xp^m = 1 = yp, yxy-1 = x1+p^(m-1) >

Note: D8=P(2,2)

Observe: |P| = pm+1, Z(P) = <xp>, |Z(P)| = pm-1, mP(P)=p2m

Theorems: m*(P) = p2m CD(P) contains p+3 subgroups

CD lattice of P(p,m)

P(p,m): How to Prove

1. Gather information about all subgroups of P

2. Find centralizers using known relations

3. 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 ofCD(P)!

P(p,m,n,r): How we found the lattice

1. Used examples and tested out patterns

2. Applied properties of p-groups and normal subgroups

3. 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 Sourceshttp://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

Any Questions?

Thank you!