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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Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
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2answers
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A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
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29 views

Representation of $A_5$

Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group $A_5$ is related to the reflection group $H_3$ or the Icosahedral group ? Thanks
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1answer
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Need help in understanding the example from Dummit & Foote text

This is an example from Dummit & Foote text. Let $D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>$.Since [$r,s$]$=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
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1answer
30 views

Index for p-adic subgroups

I always got some problems in computing precisely and understanding indexes for congruence-like subgroups. My problem seems quite simple: what is the index of $(1 + p^r)^2$ in $\mathbf{Z}_p^\times$ (I ...
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2answers
482 views

What is the correct definition of a group?

What is the correct definition of a group? More precisely the predicate "being a group"? According to Wikipedia A group is a set, G, together with an operation • (called the group law of G) that.....
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25 views

Structure of automorphism group

$\def\Aut{{\rm Aut}\,}$ Calculating the automorphism group is very hard to do. Sometimes, your group will satisfy some kind of structure which allows you to determine the automorphism group easily. I ...
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Discrete Mathematics, Equivalence Relations [closed]

I'm struggling to understand the methodology and logic in proving that a relation is reflexive. If I have $f:A \to B$ as a surjective map. How do I prove that the relation $a \sim b$ is reflexive? I ...
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2answers
579 views

H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.

Question: Let G be a group and let G' be the subgroup of G generated by the set $S=\left \{ x^{-1}y^{-1}xy \mid x,y \in G \right \}$ $\space$ Prove that G' is normal in G. Solved $\...
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Converse of Lagrange Theorem for Abelian Groups and Normal subgroup of a particular type?

Suppose $G$ is a group with $o(G) = p_1^{a_1}\cdots p_k^{a_k}$. Any divisor $d$ of $o(G)$ is of the form $p_1^{b_1}\cdots p_k^{b_k}$. Separate powers of $p_i$ and by Sylow's theorem I know their exist ...
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1answer
21 views

If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
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1answer
35 views

Related to multiplicative subgroup of positive real line

Let $F$ be a subgroup of the multiplicative group $\mathbb R^*_{>0}$ such that $F$ is dense in $\mathbb R^*_{>0}$, $$N\cap F=\emptyset\ \text{ and }\ NF=N,$$ in which $N$ is a subset of $\...
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1answer
29 views

Question on cosets and Lagrange's Theorem

If $H$ is a subgroup of $G$ and $|G| = n |H|$ where $n$ is a positive integer, how can I prove that there is some positive integer $k$ with $ 1 \le k \le n$ such that $x^k$ is in $H$ where $x$ is an ...
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0answers
21 views

Wallpaper groups

I am trying to understand the wallpaper groups and their symmetries. For example consider the following tiling : I believe that the symmetries are two translations, a $120^\circ$-rotation and a ...
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1answer
27 views

Non-simplicity of groups of order $p^{a}(p+1)$, $p$ a prime

Alperin and Bell, Groups and Representations, section 7, exerc. 10 (a), p. 71, is as follows : if $p$ is a prime number, if $G$ is a group of order $p^{a}(p+1)$, with $a > 1$, then $G$ cannot be ...
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21 views

Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
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0answers
26 views

Graph with small automorphism and large isomorphism

Is there a graph family on $n$-vertices such that any graph $G$ in the family have small automorphism group (say $|Aut(G)|\leq n^c$ for some fixed $c>0$) which if $G$ and $H$ are isomorphic then ...
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1answer
26 views

isomorphic groups related in direct product.

I want to prove that the direct product $G\times H$ of two groups has a subgroup isomorphic to $G$ and a subgroup isomorphic to $H$. How I thought to prove is that taking a pair $(g,h)$ from $G\times ...
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1answer
50 views

Is there a general strategy for identifying the automorphism group of a graph?

I understand what an automorphism is, and I can sort of wrap my head around the idea that the set of automorphisms under composition form a group, but when asked to actually find the automorphism ...
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2answers
29 views

Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$ (12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z} $$ Is this the best answer I can ...
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1answer
28 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
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1answer
46 views

Group action on cosets

I would like to solve the Problem 2.19 from A Course in Modern Mathematical Physics by Szekeres. The problem is part of the paragraph 2.6 Group action. The formulation is: Problem 2.19 If $H$ is ...
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1answer
37 views

Can someone explain this proof of the relationship between chromatic number and independence number to me?

I came across the following claim and proof in this paper, and I really don't follow. If $G$ is a vertex-transitive graph with independence number $\alpha$ and chromatic number $\chi$ then $n/α(G) ≤...
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Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
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0answers
40 views

Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
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39 views

To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$ \bigcap_{N \in \mathcal{N}} N \ = \ \{ e\} $$ I know how free ...
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1answer
34 views

Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $ S^1 \times S^1 \times S^1 $? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
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2answers
36 views

Showing $NH$ is a subgroup of $G$

Question : If $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$, prove that $NH$ is a subgroup of $G$. Thread is constructed on a mobile so I will attempt to be as succinct as possible. ...
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2answers
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A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial.

A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial, i.e., $\ f(\sigma) = 0 \ \forall \sigma \in S_{n}$. I started off by thinking that it was something to do with the ...
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1answer
42 views

Group of order $pq$ with $p<q$ prime has $q-1$ elements of order $q$

If $p<q$ are primes and $G$ is a group of order $pq$, then $G$ has exactly $q-1$ elements of order $q$. My attempt was: Use Sylow theorem to show that $G$ has only one subgroup of order $q$, so ...
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1answer
33 views

Counting maximal subgroups in a finite $p$-group

Let $G$ be a finite $p$-group. I want to show that if the number of maximal subgroups is strictly less than $p+1$ then $G$ is cyclic. This may not be true, but if the number of maximal subgroups is ...
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Does Abelianization Commute with Profinite Completion?

Let $G$ be a group, let $\widehat{G}$ be its profinite completion, and let $G^{\text{ab}}$ be its abelianization. Is is true that abelianization commutes with profinite completion, in the sense that $...
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Sifting algorithm for group generated by a set. [closed]

On page 38 of "Lecture Notes in Computer Science" by Christoph M. Hoffmann, there is an algorithm (ALGORITHM 2). I have some confusions. The algorithm needs to go to all column element indexed by ...
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Monolithic quotients in soluble groups.

Let $G$ a finite soluble group. Is it true that if $K$$\vartriangleleft$ $G$ is maximal respect to the condition $G$/$K$ non abelian, then this quotient is monolithic with monolith the derived ...
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3answers
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Showing order of $Z (G) =1$ or $pq$

Question : If $|G|=pq$ where $p$ and $q$ are primes that are not necessarily distinct. Prove that the order of $Z (G) =1$ or $pq$. Showing the order is $pq$ is trivial. I unsure how to start ...
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3answers
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$|G| = ?$ if its subgroups are $\{e\}$, $G$ itself, and a subgroup of order $7$?

Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{ e \}$, and a subgroup of order $7$. What is $|G|$? What can you say if $7$ is replaced with $p$ where $p$ is prime? I ...
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1answer
49 views

Order of a center of a group is prime order

Question : Suppose that $G$ is a non-abelian group of order $p^{3}$ where $p$ is prime and $Z (G) \neq \{e\}$. Prove that $|Z (G)| =p$. Any useful hint to this question is appreciated. ...
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Linear character and p-rationality

Let $G$ a group and $\chi$ a linear character, p$\ne$2 a prime number with $|G|$=$p^a$$m$ and $(p,m)=1$. Show that $\chi$ is p-rational if and only if p doesn't divide the order of $\chi$ (as element ...
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0answers
69 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
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On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
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Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
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1answer
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Is there a $G$-equivariant bijection $h: G/X \to G/Y$?

Let $X$ and $Y$ be subgroups of a group $G$ such that there are $G$-equivariant maps $f: G/X \to G/Y$ and $g: G/Y \to G/X$. Is there a $G$-equivariant bijection $h: G/X \to G/Y$?
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Derived graph of antipodal graph

Smith proposed a way of transforming an antipodal graph into a non-antipodal graph in the following way: Suppose $\Gamma$ is an antipodal graph, then it is imprimitive by a theorem of Smith. Then ...
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Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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Question on the quotient of a normaliser and a commutant

Let G be a group and $H\subset G$. Let $N_G(H)$ and $\text{Comm}_G(H)$ be the normaliser and commutant of H in G respectively. If the quotient group $K \equiv N_G(H)/\text{Comm}_G(H)$ is known and so ...
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How many distinct ways are there to $2$-color the $8$ vertices of a cube?

How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
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3answers
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Trying to show that any group of order four is either cyclic or isomorphic to V

I know the question has already been asked. But I have trouble with the answer. Having a non-cyclic group $\,G=\{1,a,b,c\}\,$, how can I show that $ab=c$? In my attempt, I assume that $ab=1$, and ...
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1answer
56 views

Problem in understanding some steps in proof

The problem here is about categorical construction of free groups, as in Lang's algebra (p.66-68). Theorem: For any set $S$, there exists free group $(F,f)$ determined by $S$ (here $f:S\rightarrow F$)...
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1answer
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Monomorphisms, epimorphisms and isomorphisms of groups category

I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a ...
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1answer
36 views

Infinite length Composition series

Let $G$ be a group (possibly infinite). Suppose $G$ has a composition series. I could show that any other composition series has the same length. But I cannot prove the following. Let $G \...