# Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups.

1,221 questions
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### Property of Sylow $2$-subgroups of $S_n$ and $S_{n+1}$

The question Let $P$ be a Sylow $2$-subgroup of $S_n$ and let $Q$ be a Sylow $2$-subgroup of $S_{n+1}$. Show $Q\cong P\times C_2$ iff $n\equiv 1\pmod{4}$. My attempt The first direction is ...
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### Excercise with the normal subgroup $K$ of $G$ with $K:=\bigcap\{H \text{ subgroup of } G: \forall_{x,y\in G}:xyx^{-1}y^{-1}\in H\}$

So far I have showed that $K$ is a normal subgroup and that the Operation defined on $G/K$ is abelian. Now I have to show that if $\phi:G\rightarrow A$ is a group homomorphism and $A$ is abelian....
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### How do the elements of $(G\times G) /D$ look like? $D:=\{(g,g)\in G\times G\}$ and $G$ is abelian

The first Thing I have done was to verify that $D$ is a normal subgroup. I did so by verifying that $$f:G\times G \rightarrow G, f(a,b)=a^{-1}b$$ is linear, and that the kernel of $f$ is $D$. How ...
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### Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $G\cong N \times G/N.$ [duplicate]

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$G\cong N \times G/N.$$ I tried to prove this claim, but then it seems that since $G$ is abelian then every ...
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### Order of elements from coset and Schur-Zassenhaus Theorem (Exercise $3.B.6$ FGT)

I try to solve exercise 3.B.6 from M. Isaacs's book Finite Group Theory (FGT). I give it here. Let $N \vartriangleleft G$ and $g\in G$, and suppose that when $Ng$ is viewed as an element of $G/N$,...
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### Normal subgroup and corresponding homomorphism

Topics in Algebra, a book written by Herstein, said the following thing on the first isomorphism theory: [The book says] Theorem 2.7.1 is important, for it tells us precisely what groups can be ...
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### Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$.

> Suppose that $a$ & $b$ are the generators of a free group $G$.Show that a finite generated subgroup $H$ of $G$ with index 3 exists which is not normal in $G$. The way i try to solve the ...
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### Finding a normal subgroup in $G := GL_2( \mathbb F_7)$ [closed]

For now, i know Sylow Groups and the theorem of structure for abelian groups. $G := GL_2( \mathbb F_7)$, $|G|=2^5 \cdot 3^2 \cdot 7$ I m trying to show that: There exists a normal sub group in $G$...
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### ¨Show that all the square powers of every element are in the normal subgroup of index 2 [duplicate]

If $G$ is a group and $H$ is normal subgroup of index 2 in $G$, How to show that $x^2$ is in $H$ for all $x$ in $G$? Many thanks.
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### covering spaces and equivalency of these three propositions [closed]

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ...
Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$. (a) Prove that if $xH \in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$. (b) ...