Tags » Group Theory

A solvable group that has a composition series is necessarily finite

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The fundamental theorem of geometric group theory, Part II: proof

A refresher from last week: If a group G acts on a proper geodesic metric space X properly discontinuously and cocompactly by isometries, then G is quasi-isometric to X.   912 more words


Sylow's Theorems

Sylow’s theorems in finite group theory are generalizations of Cauchy’s theorem. There are several proofs of Sylow’s theorem, but I especially like the ones that are based on the ideas of… 990 more words


McKay's Proof of Cauchy's Theorem

Cauchy’s theorem in finite group theory states that if , the order of a group , is divisible by a prime , then the group contains an element of order . 391 more words


The fundamental theorem of geometric group theory (part I), topology

I love the phrase “THE fundamental theorem of…” It’s so over the top and hyperbolic, which is unlike most mathematical writing you’ll run into.  So you know that it’s important if you run into… 1,072 more words


Intersection of Center of p-group and nontrivial normal subgroup is nontrivial

Let G be a p-group and H a nontrivial normal subgroup. Prove that is nontrivial.

Let G act on H by conjugation. Since H is a normal subgroup, this is a well-defined group action since for all . 36 more words

Group Theory

The Maths Behind the Rubik's Cube

My parents gave me a Rubik’s Cube for my tenth birthday along with a very friendly-looking guide on solving it. Not too soon afterwards, I decided to learn how to solve it. 2,560 more words