# Tagged Questions

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### Alternative to the Frattini argument

If $G$ is a finite group with $H \trianglelefteq G$, and $P$ is a Sylow $p$-subgroup of $H$, then we can show that $G = N_G(P)H$. While I'm now aware of the (admittedly much simpler/nicer) Frattini ...
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### Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
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### Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
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### A group of odd order has no non-identity elements which are conjugate to their inverse.

I want some verification for my proof to a homework problem. (Is it correct? Is there a simpler way to do this?) Let $G$ be a finite group of odd order and suppose there is an element $g$ that is ...
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### $S_n$ acting on $\{1\;…\;n\}\times \{1\;…\;n\}$

Let $X=\{1,\;...\;n\}$ and $S_n$ act transitively on $X\times X$ i.e. $s:\;(m,n)\mapsto (s(m),s(n))$. Compute the orbits under this action. Attempt: I claim that there are only two orbits, ...
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### Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
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### Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$

Claim: Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$ May I know if my proof is correct? Thank you. Lemma: $G/Z(G)$ is ...
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### Some problems in group theory

May I know if my proof/solution is correct? Thank you v. much. 1.) If $G, H$ are finite groups of order $10$ and $21$ respectively, then every homomorphism $f:G \to H$ satisfies $f[G] = \{e_H\}.$ ...
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### If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$ May I know if my proof is correct? Thank you v. much. Proof: |G:(H \cap K)| \leq |G:H||G:K| ...