Exercises for group actions?

Asked Dec 30 '19 at 19:04

Active Dec 30 '19 at 19:10

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I am reading Dummit and Foote and I feel like I am weak with group actions. I would very much like to have more practice regarding them, but none of the books I have seen go into group actions as much as D&F, and I haven't seen enough exercises in D&F of the type I am looking for. I would like to do more problems of the sort:

1) Prove that if $Q_8$ acts faithfully on a set $S$, then $|S| \ge 8$.

2) If $G$ is a group of order $12$ which has four 3-Sylows, then $G \cong A_4$.

3) Let $G$ be a $p$ - group which acts on a set $S$, and let $S^G$ be the set of fixed elements of the action. Then $|S| \equiv |S^G| \pmod p$

Is there any source of these, or could you please tell me more of these problems? Thank you so much!

edited Dec 30 '19 at 19:10

asked Dec 30 '19 at 19:04

Blue

1

All three questions have been answered here. For example, see this duplicate for the second question. So the source is this site here (or a book on group theory). Can you find the other two questions? (Hint for 1. here). – Dietrich Burde Dec 30 '19 at 19:13
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Google for "group actions" and "exercises" to find more! – Dietrich Burde Dec 30 '19 at 19:18
Usually Sylow theorems are proved using group actions, so looking at that section might yield some exercises. – Justin Young Dec 30 '19 at 20:58
@DietrichBurde Thanks, I will look around! – Blue Dec 31 '19 at 00:05
You can start from MSE itself by tagging "group-actions" – Dec 31 '19 at 08:39

Exercises for group actions?

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