# 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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### Groups with order divisible by $d$ and no element of order $d$

It occurred to me that I somehow believe the following statement without actually knowing how to prove it: for every composite natural number $d$ there is a group whose order is divisible by $d$ yet ...
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### Character theory questions

I am following the text by Isaacs on character theory and I have a few questions. From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a ...
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### All subgroups of a finite abelian p-group

Given a finite abelian p-group: $G = \displaystyle\prod_{i=1}^n p^{k_i}\mathbb{Z}_{p^k}$ for some integers $k,k_1,...,k_n$. Regarding elements of G as tuples $(x_1,...,x_n) \in G$, I can get ...
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### A proof of Sylow theorem

Having proved the Sylow theorem for general linear group over finite field, how to prove it for any finite group?
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### Presentation of Borel subgroup of GL(2,p)

The Borel subgroup $B$ of GL(2,p) is the subgroup group of upper triangular matrices. It is easy to see that it is (internal) semi-direct product of two subgroups: $B=U\rtimes T$ where $U$ is the ...
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### Isomorphism between $I_G/I_G^2$ and $G/G'$

Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it. $G$ is ...
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### Non isomorphic groups who product with Z is isomorphic [duplicate]

Are there groups $G$ and $H$ such that $G$ and $H$ are not isomorphic but $G \times \mathbb Z$ and $H \times \mathbb Z$ are?
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### Reference / Survey article on automorphisms of groups

can one suggest a survey article on automorphisms of $p$ groups, and automorphisms of abelian groups/ abelian $p$ groups?
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### The Frattini subgroup contains a power of a certain subgroup

This is from an article by Hall from 1961; it's probably one of the most trivial observations in that article, but I can't get the reasoning. Let G be some group and let A be a normal abelian p-...
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### Residual Finiteness of Fundamental Groups of Seifert Fibered Spaces

I'm trying to understand why, if $S$ is a Seifert fibered space, then $\pi_1(S)$ is residually finite. From theorems 12.2 and 11.10 in Hempel's "3-manifolds", we can work with a finite-sheeted ...
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### Homomorphism of Groups and divisibility of orders

I've been studying for my exams and bumped across this one particular question that I've been having a tad difficulty on: Suppose that $\phi$ is a homomorphism from a finite group $G$ onto $H$ and ...
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### How to prove a permutation group of degree n can be generated by (at most) n-1 elements?

Given a subgroup G of $S_n$. I want to prove that G can be generated by (at most) n-1 elements. All my ideas so far seem irrelevant. A hint would be greatly appreciated. My attempts so far: n-1 ...
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### Symmetries of Cube

The group of orientation preserving isometries of Cube is $S_4$. But if we allow orientation reversing isometries also, then the group will be of order 48. What is this group (Structure)? ( Part of ...
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### Subgroups whose order is relatively prime to the index of another subgroup

Suppose that $H, K$ are subgroups of a finite group $G$, with $|H|$ relatively prime to $|G:K|$. Does it necessarily follow that $H \leq K$, or is there a counterexample? This question arose from ...
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### An 'opposite' or 'dual' group?

Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried ...
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### subgroup of $\mathbf{Z}_9$ generated by $8$

If I find the elements generated by $8$, can I say that the set of these elements is a subgroup of $\mathbf{Z}_9$ provided all these elements are in $\mathbf{Z}_9$? or to show that the set of these ...
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### Isomorphic groups

I know that there is a formal definition isomorphism but for the purpose of this homework questions, I call two groups isomorphic if they have the same structure, that is group table for one can be ...
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### A question about cyclic groups

How does one see that the cyclic group $C_n$ of order $n$ has $\phi(d)$ elements of order $d$ for each divisor $d$ of $n$? (where $\phi(d)$ is the Euler totient function)
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### A torsion-free quotient?

Let $G$ be a group and $T$ the set of elements of finite order in $G$. If $T$ is a subgroup of $G$, then $G/T$ is a torsion-free group. Suppose $G$ is a compact Hausdorff topological group. Is it ...
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### Proof that a group action is primitive in terms of point stabilisers

Can someone help me with proof of the next statement? Suppose that the group $G$ acts transitively on the set $A$, and let $H$ be the stabiliser of $a\in A$.Then $G$ acts primitively on $A$ if and ...
Can we embedd $GL(2n,q)$ into $GL(n,q^2)$ for $n\in \mathbb{N}$ and $q=p^m$, $p$ a prime? If yes, how?
Let $G=\{a_2, a_2, a_3...a_n\}$ be a finite abelian group of order $n$. Let $x=a_1 a_2 a_3...a_n$. Prove that $x^2=e$. My solution: Well since the order of the group is finite, we know the order of ...