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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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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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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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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18 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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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
48 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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are there $E(x,G)$ and $E(H,G)$ two subgroups of $G$? [closed]

let $G$ be a group and $H$ be a subgroup of $G$. let $E(x,G) = \{g \in G ; [g,x,x]=1 \}$ and $ E(H,G) = \{ g \in G ; [g,x,x] = 1 ~~~ \forall x \in H \}$ . are there $E(x,G)$ and $E(H,G)$ two ...
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27 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
35 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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11answers
2k views

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 ...
2
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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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0answers
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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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
45 views

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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55 views

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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35 views

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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31 views

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
30 views

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
55 views

$|G| = ?$ 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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20 views

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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67 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}...
4
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0answers
47 views

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
59 views

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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15 views

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 ...
4
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2answers
64 views

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
25 views

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 ...
1
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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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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 \...
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2answers
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Cardinality of a vector space over a finite field [closed]

Let $V$ be a vector space over $\mathbb{Z}_5$ of dimension $3$. What is the cardinality of $V$? I don't know how to proceed. Thanks in advance.
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Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
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1answer
34 views

Showing a group is not the internal direct product of cyclic group

Question: Let$ G={3^{a}6^{b}10^{c}}$ under multiplication and $H={3^{a}6^{b}12^{c}}$ For all $a,b,c \in R$ Prove that$ G=<3>x <6>x <10> $ and H is not $<3>x <6>x &...
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1answer
21 views

How to save a pc-group in MAGMA?

I have a group $M$, which is a large matrix group. Using the command $\texttt{LMGSolubleRadical(M)}$ I obtained $R\cong M$, where $R$ is of type $\texttt{Grppc}$, and an isomorphism $M\to R$ called $\...
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1answer
22 views

Distinct coset representative and stabilizing an element.

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
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0answers
28 views

Number of elements of Complete Right Transveral in pointwise stabilizer. [closed]

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
1
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1answer
27 views

Schur Multipliers in Finite Simple Groups

I heard that Schur multiplier's played important role in classification of finite simple groups. By means of simple example, can one illustrate how the Schur multiplies played their role in the ...
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1answer
20 views

Relationship between elements in an internal direct product

Question: Let $H$ and $K$ be subgroups of a group $G$. If $G=HK$ and $g=h\bar {k} $, where $h\in G $ and $\bar {k} $. Is there any relationship among $|g|$, $|h|$ and $|\bar {k}|$? What if $...
2
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1answer
33 views

Properties of group extensions

There is an excercise in Rotman's Introduction to the theory of groups: which of the following properties, when enjoyed by both $K$ and $Q$, is also enjoyed by every extension of $K$ by $Q$? finite $...
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1answer
16 views

Express a group in four different direct product

Question: Express U (165) as an internal dirext product of proper subgroup in four different ways. The hint suggests the use of the Chinese remainder theorem which I am unfamiliar with. Is there ...