1 / 26

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

On the Chermak-Delgado Lattices of Split Metacyclic p-Groups. Research by B rianne Power, E rin Brush, and K endra Johnson-Tesch. Supervised by Jill Dietz at St. Olaf College. Background. Chermak and Delgado (1989) were interested in finding families of characteristic subgroups. They

Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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

  2. 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

  3. 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 }

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

  5. 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.

  6. 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.

  7. 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

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

  9. (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

  10. 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

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

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

  13. 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)

  14. 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

  15. CD lattice of P(p,m)

  16. 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

  17. 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) >

  18. 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)

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

  20. 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)!

  21. 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

  22. 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

  23. 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.

  24. 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

  25. Any Questions?

  26. Thank you!

More Related