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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Complement of Normal subgroups and free groups

Does every normal subgroup has complement in free groups? What about free abelian groups i.e. Is free abelian gorup complemented group? Definition: If there exist a subgroup K such that HK = G and H ∩...
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$G$ nilpotent group and $N\trianglelefteq G$ then $[N,G]<N$, attempt of the proof

I need help in proving this fact: Let be $G$ nilpotent group and $N$ a normal and non trivial subgroup. Then $[N,G]$ is a proper subgroup of $N$. My attempt: I know the following fact: Let be $H$ ...
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Suppose $H$ is a subgroup of $S_n$ then does there exist $i\in \{1,2…,n\}$ such that $H=\mathbb{Stab}(i)$?

Suppose $H$ is a subgroup of order $(n-1)!$ in $S_n$ then does there exist $i\in \{1,2...,n\}$ such that $H=\mathbb{Stab}(i)$ ? My motivation behind asking this question comes from a question on ...
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$\forall x \in Fix( \sigma ),\ \mathcal{O}(x)=\{ x\} $ and $\forall x \in supp( \sigma ), \{ x\}\subset \mathcal{O}(x)$

Let $\sigma \in \mathfrak{S}_{n}$ Show that : $$\forall x \in \operatorname{Fix}( \sigma ),\ \mathcal{O}(x)=\{ x\} \quad \rm{ and }\quad \forall x \in \operatorname{supp}( \sigma ),\ \{ x\}\...
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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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36 views

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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Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
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61 views

Using Burnside's Lemma in GAP to handle special variations of the Rubik's Cube?

If you want to count the number of distinct positions of a standard 2x2x2 Rubik's Cube simple counting arguments will suffice: There are 8 corners, all distinct The 8 corners can be in any ...
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infinite Dihedral group is unimodular

How to show that the infinite Dihedral group is unimodular? A locally compact group G is strongly amenable if, for every open relatively compact neighborhood C of the unit which is symmetric we have ...
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All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
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Hall System of a finite solvable group

Let $G$ be a finite solvable group and $\pi(G)$ be the prime divisors of $G$. For each $p\in \pi(G)$, let $S$ be the Sylow $p$-complement of $G$ If $\pi\subseteq \pi(G)$, let $\pi^* = \pi(G) \...
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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. ...
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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: $\alpha^5$=...
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Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
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Is there a theorem stating that disjoint cycles generate distinct elements?

If we have a group $H=\langle (12345),(678) \rangle$, it's obvious that $|H|=|(12345)|\cdot |(678)|=15$, because the cycles are disjoint. Is there some theorem stating this?
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Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
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Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
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quotient group of finite fields

I could not define the elements of the quotient group $F^*_{q^{d}} / F^*_{q}$. Let $F^*_{2^{4}}$={$1, a, a^2, a^3, a^4=a+1, a^5=a^2+a,...a^{14}=a^3+1,a^{15}=1$} s.t. $ a^4+a+1=0$ and $F^...
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Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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31 views

Proof of a theorem on Reflection Groups

I am reading the book Finite Reflection Groups by Grove and Benson. I didn't understand the following proof. See $(a_1,t)$. What is $t$ here? Then Why the inequality $(r,t)-2(r,r_{i_1})(r_{i_1},t)&...
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Class Preserving Autmorphisms [closed]

What are Camina p groups? What special properties do Camina groups of Class 2 have over those of class 3?
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Groups with order a product of unrelated distinct primes

Consider a group of size $n$, where $n$ is the product of distinct unrelated primes (two primes $p$ and $q$ are unrelated if $q \nmid (p-1)$ and $p \nmid (q-1)$). The claim is that there is only one ...
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If $|G/H|=4$ then $G$ is union of three proper subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three ...
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1answer
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Minimum number of elements in group

Suppose we have $G=\langle(123),(456),(14)(25)(36)\rangle$ a subgroup of $S_6$ Am I correct in saying that the minimum number of elements in $G$ equals the product of the orders of the elements of ...
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Irreducible action of a group

Suppose that $G = HN$ is a finite solvable group where $N$ is a minimal normal subgroup of $G$ Does $H$ act irreducibly on $N$? I know that $N$ is an elementary abelian $p$-group of $G$. I need to ...
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“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
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Show that the centralizer is generated by these $3$ elements

$\sigma = (123)(456)$. Show that $C_{S_6}(\sigma)=\langle(123),(456),(14)(25)(36)\rangle$. So we know that there are $40$ elements conjugate with $(123)(456)$ in $S_6$. Then it follows that $|C_{...
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Number of elements set of generators

Suppose $G$ is a subgroup of $S_6$ generated by $G=<(123),(456),(14)(25)(36)>$. Is it safe to assume that products of distinct generating elements are also distinct? What I mean is, can we be ...
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1answer
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reference for classifying groups of order $p^2q^2$

In a previous question I asked about the number and structure of groups of order $p^2q^2$ where $p,q$ are primes and with the help of Prof. Derek Holt I understand it now (see here non-abelian groups ...
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What is the centralizer of $(123)(456)$ in $S_6$?

Given that $\sigma = (123)(456)$. Compute $C_{S_6}(\sigma)$ (instead of writing out all the elements, write down the elements that generate the centralizer). If $\sigma$ were an $m$-cycle we chould ...
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The representation of $SU(2)$ as a polynomial function on $\mathbb C^2$

Let $A$ element of $SU(2)$ and $p$ a polynomial function of fixed degree $l$ on $\mathbb C^2$ (in other words, $p \in P_l(\mathbb C^2)$), then the polynomial representation of $A$ in $P_l(\mathbb C^2)$...
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Group of order $54$ has normal sugroup of order $27.$

Let $G$ be a group of order $54$. Prove that there exists a normal subgroup of order $27.$ Is this normal subgroup unique? Thoughts. Since $27$ divides $54$, by Lagrange's theorem we can not exclude ...
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G acts on X transitively, then there exists some element that does not have any fixed points

Let $X$ be a transitive $G$-set. ($G$ acts on $X$ transitively.) If $X$ is finite and has at least two elements, show that there is some element $g$ $\in$ G which does not have any fixed points; that ...
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The number of different G-actions on X [closed]

Let $X$ $=$ $\{$$1$, $2$, $3$$\}$ and $G$ $=$ $\mathbb Z_2$. How many different G-actions are there on $X$? Just learned group action. Need some hint on this one. Thanks.
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Number of elements in $S_6$ conjugate to $(123)(456)$

Find the number of elements in $S_6$ conjugate to $(123)(456)$ I know we're only looking at elements in $S_6$ with the same cycle type as $(123)(456)$ (two 3-cycles). So we have the following: $$\...
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Dirchlet region for the Hecke Triangle group

Let $G_n$ for $n>2$ be the subgroup of $SL_2(\Bbb R)$ generated by $$ \begin{bmatrix} 0 & -1\\ 1 & 0 \\ \end{bmatrix} \ \text{and} \ \begin{...
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1answer
45 views

Classifying groups of order 6

I'm trying to proof that if a group $G$ has order $6$, then it is either $\mathbb{Z}_{6}$ or $S_{3}$. I know that there are a lot of solutions to this on the internet, but I want to know why I found ...
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Abelian fitting subgroup of a solvable groups

Let $G$ be a finite solvable groups and $F(G)$ be a Sylow $p$-subgroup of $G$, for some prime $p$ such that $F(G)$ is Abelian. I want to know what things we can say about $F(G)$ in this case?
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Subgroup of $\left(GL_2\left(R\right),\:\cdot \right)$

Is $t\left(GL_2\left(\mathbb{R}\right)\right)=\left\{x\in GL_2\left(\mathbb{R}\right)|\:ord\left(x\right)\:<\:\infty \right\}$ a subgroup of $\left(GL_2\left(\mathbb{R}\right),\:\cdot \right)$ ? ...
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Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$

Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$ My attempt We begin to find all elements of order $4$ in $Z_4 \oplus Z_4.$ First attempt is to find all the cyclic subgroups of order $4.$ We want ...