# Tagged Questions

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### Group acting on a set.

Let $G$ be a group of order $7$ acting on a set of $5$ elements. Show that the action of $G$ must have a fixed point.
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### When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$. Thanks for your help.
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### Without using Sylow: Group of order 28 has a normal subgroup of order 7

Prove that a group of order 28 has a normal subgroup of order 7. How can I prove this without using Sylow's theorem? I know by Cauchy’s theorem, there exists an $x\in G$ with order 7, now I just ...
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### Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
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### Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
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### Classification of transitive G-sets for a given group of small order

Given a group of small order (<30), how does one go about systematically finding all the transitive G-sets up to isomorphism? By X and Y being isomorphic we mean there are maps $f:X \rightarrow Y$ ...
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### Finding the kernel of an action on conjugate subgroups

I'm trying to solve the following problem: Let $G$ be a group of order 12. Assume the 3-Sylow subgroups of $G$ are not normal. Prove that $G\cong A_4$. Here's my attempt: let $\mathscr S$ be ...
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### Diagonal groups and semidirect products

I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner ...
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### trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
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### Rotman's Introduction to to the theory of groups. Exercise 3.45.

Can you give me a hint on the first part of the exercise? Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
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### Different actions of an affine primitive group?

Fairly new to group actions and I'm having trouble finding answers to these in textbooks... Say we have a primitive action of $G$ on $\Omega$, with regular elementary abelian socle $N$. Now suppose ...