# Tagged Questions

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 studies groups.

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### Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
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### Normalizer of $H_0 \times H_m$ in $S_{2n}$

Let $H_0 \subsetneq \dotso \subsetneq H_m$ be a chain of subgroups of $S_n$ (the symmetric group of $n$ elements) such that holds: $N(H_i) = H_{i+1}$ and $N(H_m) = H_m$. I want to prove that the ...
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### Number of elements not equal to their inverses is even number

In any finite group, number of elements not equal to their own inverses is even number In my book they have paired elements with their inverses, being elements and inverses different from each other. ...
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### Group theory question proving associativity

I am doing this group theory question: I have already proven that * is commutative, however, I'm I bit confused about proving for associativity. I used three variables a, b and c and said: RTF ...
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### Necessary and sufficient condition for a normal group to be kernel of a homomorphism from the group to itself

I am looking for a necessary and sufficient condition for a subgroup $K$ of a group $G$ to be kernel of a homomorphism $\phi$ from $G$ to $G$. The tools that come into my mind is first isomorphism ...
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### By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
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### Restricted linear representations of abelian groups

If $G$ is a group (say finite for simplicity though the question applies to infinite groups as well), what can one say about the subgroup $G^*_n = \text{Hom}(G, \mu_n)$ of the group of all linear ...
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### Why are these groups isomorphic?

I have this group of permutations: And I have this group of complex numbers: These groups are isomorphic to each other, but it seems I do not understand why. I was looking for similarities in ...
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### Which vertex-transitive planar graphs represent non-self-intersecting polyhedra?

Consider an infinite planar graph with the following properties. Its vertices all have valence $3$. The faces all have $5$ edges. Now put it in cartesian space and require that the faces are all ...
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### Action of $Aut(UT_3({\mathbb{F}_p}))$ on the set of non-commuting elements of $UT_3({\mathbb{F}_p})$

Assume that $G$ is the group of 3x3 unitriangular matrices over the field of $p$-elements $\mathbb{F}_p$. Furthermore, assume that the group of automorphisms of that group, $Aut(G)$, acts on $G$. Do ...
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### Finite groups and prime divisors. Understanding how to deduce a claim from a certain proof.

In my algebra textbook, it goes like this. First, there is presented Cauchy's theorem: Let $G$ be a finite group, and let $p$ be a prime divisor of $|G|$. Then $G$ contains an element of order $p$....
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### Prove that following set is a group [duplicate]

Given $\mathbb{R} \setminus \{-1\}$ with the operation $a*b=a+b+ab$, check whether is it or not a group. My solution: Associative $$(a*b)*c=(a+b+ab)*c=a+b+ab+c+(a+b+ab)c=\\ =a+b+ab+c+ac+bc+abc;$$ ...
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### Subgroup of Index 2 is Normal - If $g \not \in H$, then $gH= G/H$?

I'm trying to understand the following proof: https://proofwiki.org/wiki/Subgroup_of_Index_2_is_Normal Specifically the part: If $g\not\in H$, then $gH=G/H$ as there are only two cosets and the ...
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### A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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### Is Cayley's theorem tight in a sense? [duplicate]

For any $n$, does there exist a group $G$ of order $n$ so that $G$ is not isomorphic to any subgroup of $S_j$ for $j<n$?
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### If $G$ is cyclic, then $G=\{a^0, a^1, …\}$, but why does it have to be that $a^k=e$ for some $k$?

If $G$ is cyclic, then $G=\{a^0, a^1, ...\}$, but why does it have to be that $a^k=e$ for some $k$? I.e. that for $G$ to be cyclic, then surely the generating element would need to generate $e$ also. ...
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### How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
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### Show that there has to be an orbit with at least $3$ elements
Let $H \leq A_4$ be a subgroup of $6$ elements. $H$ acts on $X=\{1,2,3,4\}$ by $\sigma \cdot i = \sigma(i)$. Show that for $H$ to act on $X$ there has to be an orbit with at least $3$ elements. ...
### Order of subgroup generated by two cyclic subgroups in $S_6$.
Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...