# Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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### Conjugacy classes of a group of order $8k$

Let G be a group of order $8k$, show that there are at least 5 different conjugacy classes. Hi everyone, I have this problem I think I had a solution involving stabilizers, however I feel there must ...
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### $H\lhd G$, $|H|=p^k$ then $H$ is included in every $p$-Sylow subgroup of $G$

Let $G$ be a finite group and let $H$ be a normal subgroup of G, with $|H|=p^k$. Then you have to prove that $H \subset P$, for every $P\in\mathrm{Syl}_p(G)$. What I thought is that, as $p$-Sylow are ...
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### Are the only single element normal subgroups precisely the elements that make up the center of a group?

So, a single element normal subgroup $n$ of group $G$ would be defined as $\forall g \in G: gng^{-1} = n$. All elements of an abelian group would qualify, and so would more generally the elements of ...
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### Transitive action of a discrete group on a compact space

Let $G$ be a discrete countable group acting on a compact, Hausdorff space $X$. Assume that the action is transitive. Namely, $G\cdot x=X$, for all $x\in X$. Does it follow that $X$ is finite? I ...
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### On the Sylow subgroups of a subgroup

What I am trying to prove: Let $G$ be a finite group and $H$ a subgroup of $G$. Then prove that two $p$-sylow of $H$ are contained in different $p$-sylow of $G$. My attempt was suppose that they are ...
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### Reference request: structure of stabilisers of totally isotropic subspaces in orthogonal (and unitary) groups

I am looking for a book or paper which covers the structure of stabilisers in $GO(n,F)$, $SO(n,F)$ (or maybe in $\Omega(n,F)$) of totally isotropic subspaces of dimension $k$. Can you please suggest ...
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### Non-abelian finite $p$-group in which every element has prime order? [duplicate]

Let $G$ be a finite group such that for some prime $p$, every element has order $p$ . Can $G$ be non-abelian ? My thoughts: Of-course $G$ is a $p$-group so has non-trivial center. If $G$ is non-...
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### Sylow subgroup of a subgroup 5

I am aiming for looking for all the Sylow subgroups of classical groups, which gives me a seemingly elementary question: what can we say about Sylow subgroups of a subgroup if we know all Sylow ...
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### Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group

Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group I'm looking for a quick example of this, I've been trying to figure ...
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### How to show $\mathrm{Hom}_\mathsf{Ab}(\mathbb Z/a\mathbb Z,\mathbb Z/b\mathbb Z) \cong \mathbb Z/(a,b)\mathbb Z$?

With some help, I figured out how to show $\mathrm{Hom}_{R-\mathsf{Mod}}(R/I, N) \cong \{n \in N \mid \forall a \in I, an=0\}$. I would like to use this result for the current problem. We can ...
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### What are some good books on group theory? [duplicate]

I want to self-study group theory. Does anyone have good recommendations for me?
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### If $A\oplus B\approx A$ then $B=0$? [duplicate]

Suppose $A,B$ abelian groups, such that $A\oplus B\approx A$, can I conclude that $B=0$? If it's true, is there any hint how to prove it?
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### Help with disjoint cycle decomposition and finding image/pre-image of a S8 group

i have this exercise and i'm confused about the part where i find the image and pre-image. I solved it for 5 in alfpha, found the image and preimage (preimage is number that 5 results from and image ...
According to wiki, Isomorphism: Given two groups $(G, ∗)$ and $(H, \odot }$, a group isomorphism from $(G, ∗)$ to $(H, \odot }$ is a bijective group homomorphism ...
### Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram
I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$: First definition Given $\sigma \in S_n$ ...