A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

learn more… | top users | synonyms (2)

4
votes
2answers
67 views
+50

$G$ finite, the number of distinct conjugates of $x$ is the index of the normalizer $N_x$ of $\{x\}$ in $G$

In order to prove this, I did the following: first, I showed that conjugacy forms an equivalence relation, then I can find its conjugacy classes. I understand how to form a conjugacy class, given a ...
0
votes
0answers
73 views
+50

How many non isomorphic semidirect products are there between $\mathbb Z_2$ and $SL(2,3)$?

I know that $GL(2,3)$ is one of this, but i need the characterization of all possibles of the semidirect products between $\mathbb Z_2$ and $SL(2,3)$. Thanks, for any help.
8
votes
2answers
126 views
+50

Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...