Tags » Group Theory

The Cauchy-Schwarz inequality and the Lagrange identity

The classical Lagrange identity is the following:

This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part. 288 more words


Cauchy's Theorem

The famous Lagrange’s Theorem states that, in any finite group , the order of any element divides the order of . A natural follow up to this theorem is whether or not the converse holds. 433 more words

Group Theory

Bonus Haiku: A Late-Night Group Theory Haiku.

I wrote this one Mar 9 at 12:21am whilst finishing homework. Like I am usually doing when genius strikes.

Is plus-minus one,
Center of quaternions?
Thus normal subgroup?


Conjugacy classes in A_n

We know that two permutations in are conjugate if and only if their decompositions consist of the same cycle type. And a conjugacy class in of even permutations is either equal to a single conjugacy class in , or splits into two conjugacy classes in . 471 more words

Alternating Group

Computing the sign of a permutation

In this post we’ll go through the exercises of the last. It will therefore mostly be examples based. Let’s recap. We introduced the sign function. We found that was generated by 2-cycles. 527 more words

Group Theory

The sign function of the symmetric group

Our aim in this post is to decide whether there is an epimorphism from to . Obviously we need . Also , so we might as well assume . 1,159 more words

Group Theory

More on homomorphisms

Recall that a homomorphism was a structure preserving function that need not be injective or surjective (unlike an isomorphism). Structure preserving is completely captured algebraically by the identity holding for all . 820 more words

Group Theory