Duncan Bennett

In mathematics, we may sometimes take "unique factorization" for granted. For example, for any natural number n, we can factor n into a product of unique primes of unique powers. Conversely, if given a collection of primes of unique powers, we will get a unique natural number. However, if we replace natural numbers with groups and primes with simple groups, the converse is not necessarily true. 

The Jordan Holder theorem allows one to "factor" finite groups uniquely into simple groups. This gives rise to the following question: Given two simple groups N and K, what are the groups G such that G/N \cong K? This is known as the extension problem in algebra. This talk will present how results from group cohomology apply to the extension problem for finite groups.