# Questions tagged [normal-subgroups]

For questions concerning normal subgroups of groups. Consider using with the (group-theory) and/or the (abstract-algebra) tags too.

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### Show that $H$ is a normal subgroup of $G.$

This question was asked in my mock test of masters entrance test and I couldn't prove one of the question: Question $\to$ Let $G$ be a group of order $105$ and $H$ be it's subgroup of order $35$. ...
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### Conjugacy classes of normal subgroup in group

Let $G$ be a finite group, and $N$ be a normal subgroup of index $p$. If every conjugacy class of $N$ is also a conjugacy class in $G$, what can we say about $G$ or $N$? Such instances occur if $G$ is ...
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### Suppose $H$ is a normal subgroup of $G$ with prime index $p$, does this implies that $H$ contains $G^p$?

The power subgroup is defined as $G^p =\left\{g^p \mid g \in G\right\}$. Question: If $H$ is a normal subgroup of $G$ with prime index $p$, does this imply that $H$ contains $G^p$? This statement is ...
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### Can there be a subgroup $K$ of a group $G$ such that for some $a \in G$, $aK \subseteq Ka$ but $Ka \nsubseteq aK$? [duplicate]

I have a question, in a sense, about how asymmetric left and right cosets can be when dealing with an infinite, non-normal subgroup $K$ of a (non-abelian) group $G$. Specifically, my question is ...
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### Groups of order $2^n p$ for $n\geq 1$ and $p$ prime with $2^n> (p-1)!$ are non-simple. Is my proof correct?

I'm doing my homework in Group Theory and as part of an exercise, I want to show the following Lemma: Let $n\geq 1$, $p$ a prime, s.t. $2^n > (p-1)!$ and $G$ a group of order $2^n p$. Then G has a ...
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### The free product has the direct product as a factor group. What's the corresponding normal subgroup?

Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
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### Abelian + normal $\implies$ central?
Here is the question, it is from a 2023 PhD qualifying exam at my university: Let $G$ be a finite group and $H$ a normal subgroup of $G$ of order 5. Prove that if $H$ contains an element not in the ...