Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose $G$ is a finite group with conjugacy classes $C_1,C_2,\dots,C_\ell$. Suppose we take one element from each conjugacy class: $g_i \in C_i$ for all $i=1,\dots,\ell$.

Is it true that $G = \langle g_1,g_2,\dots,g_\ell \rangle$ (i.e. $G$ is generated by these elements)?

If this is true, references? Hard to prove?


Edit: Thanks again everyone! I guess I should have looked around more on overflow first :)

share|cite|improve this question
up vote 5 down vote accepted

This was asked, and answered, on MathOverflow some time ago:

share|cite|improve this answer

Yes. Suppose not: then there will be some maximal subgroup $M\le G$ intersecting each conjugacy class. Then, because $G$ is the union of its conjugacy classes, $G$ is the union of conjugates of $M$. But this is impossible. (Can you see why? Try counting how many elements one can have in the union of $M$ and all its conjugates, noting there are at most $[G:M]$ such conjugates.)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.