CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
The difference between isomorphism and non-isomorphism James Wilson, CSU
October 18, 2011 Math 350 2-3pm	Abstract Deciding that two groups are isomorphic is a clear task: simply provide an invertible homomorphism between the groups. On the other-hand, understanding why two groups are non-isomorphic takes many different forms. There is a family of groups of size $N$ with nearly exponential numbers of isomorphism types. Yet, these groups are virtually indistinguishable: they have the same character tables, all centralizers of non-central elements are identical, their automorphism groups break into at most 2 flavors and are represented isomorphically on the abelianizations of the groups, and several other typically discerning isomorphism invariants fail to help understand these groups. Surprisingly, despite all these similarities, it is possible to test if a two of such groups are isomorphic using only a logarithmic number of steps (compared to $N$).

The difference between isomorphism and non-isomorphism

James Wilson, CSU

October 18, 2011

Math 350

2-3pm

Abstract

Deciding that two groups are isomorphic is a clear task: simply provide an invertible homomorphism between the groups. On the other-hand, understanding why two groups are non-isomorphic takes many different forms. There is a family of groups of size $N$ with nearly exponential numbers of isomorphism types. Yet, these groups are virtually indistinguishable: they have the same character tables, all centralizers of non-central elements are identical, their automorphism groups break into at most 2 flavors and are represented isomorphically on the abelianizations of the groups, and several other typically discerning isomorphism invariants fail to help understand these groups. Surprisingly, despite all these similarities, it is possible to test if a two of such groups are isomorphic using only a logarithmic number of steps (compared to $N$).