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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How to prove that G is a cyclic Group?

Suppose $G$ is a finite Abelian group and, $\forall n\in \mathbb{N} $, there exist less than $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
3
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1answer
20 views

When can you “extend” subgroups?

Suppose $H\le G\le G'$ are subgroups, with $H$ finite index in $G$. Can you always find a subgroup $H'\le G'$ of finite index such that $H'\cap G = H$? Of course it's trivial if $G$ is finite index ...
-1
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0answers
19 views

$ G $ is soluble, then every properties of $ G $ is inherited by $ G/N $ ?

Let $ G $ be a finite group and $ G $ is soluble. Suppose $ N $ be a normal minimal subgroup of $ G $. Then every properties of $ G $ is inherited by $ G/N $ ?
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0answers
76 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
7
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0answers
22 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
4
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2answers
23 views

$G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; $G$ is not simple

Let $G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; then how to show that $G$ is not simple ? I have proceeded by Cayley's theorem , $\ker f$ is ...
2
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0answers
15 views

Frattini subgroups and nilpotent groups: bijection? [duplicate]

I have been proved that the Frattini subgroup of a finite group is nilpotent. Now I am wondering: is the converse true? I mean, if $G$ is a finite nilpotent group, is there always a finite group ...
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0answers
35 views

I need the paper “ J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365” [on hold]

I need this paper " J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365", But i not found. Who can help me to find this paper ?
3
votes
2answers
53 views

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $\mathbb Z$-module for $m < n$?

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $ \mathbb Z$-module for $m < n$ ? I think it's not possible. It might be a easy problem for some of you, but I really don't ...
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1answer
16 views

Definition of $ p $-supersoluble group.

I was searching for definition of $ p $-supersoluble group but not find definition. Please help me.
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1answer
15 views

$ K/N $ be a normal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? [on hold]

Let $ G $ be a finite group and $ N $ is a normal subgroup of $ G $. Suppose $ K/N $ be a maximal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? ...
1
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1answer
20 views

raising elements of profinite groups to $p$-adic powers

Let $\widehat{F_2}$ be the profinite free group of rank 2, and let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and $\widehat{\mathbb{Z}}^\times$ its group of units. For ...
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3answers
64 views

Find the subgroup of $GL(2,\mathbb{C})$ generated by two matrices $A$ and $B$.

Find the subgroup of $GL(2,\mathbb{C})$ generated by the matrices $A$ and $B$, where $A=\begin{pmatrix} 1 & 0\\ 0 & i \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1\\ -1 & 0 ...
0
votes
1answer
40 views

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$.

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$. Where $V_4$ is Kleins 4 group and $\mathbb{C}^{*}$ is multiplicative group of nonzero complex numbers.
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0answers
32 views

What is the relation between $\mathcal{G}_{12}, \mathcal{G}_{23}, \mathcal{G}_{13}$?

please help me to find this. Suppose that the number of group homomorphism from the group $G_1$ to $G_2$ is $\mathcal{G}_{12}$, from $G_2$ to $G_3$ it is $\mathcal{G}_{23}$ and from $G_1$ to $G_3$ ...
9
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3answers
56 views

Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

What is the smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi-direct product of cyclic groups? So finite abelian groups are ruled ...
1
vote
1answer
11 views

Let $ G $ satisfying the maximal permutizer condition, then $ G/N $ satisfying the maximal permutizer condition ?

Let $ H $ be a proper subgroup of finite group $ G $. Then permutizer $ H $ in $ G $ is defined by $ P_{G}(H) = \langle y \in G \vert \langle y \rangle H = H \langle y \rangle \rangle $. A group $ G $ ...
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0answers
21 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
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1answer
21 views

Significance of module in groups [on hold]

What is the significance of module in groups . I mean it is used for participation of sets but what is the its use in groups.
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0answers
35 views

On the number of group homomorphisms from $S_n$ to $S_m$

I was studying the number of group homomorphisms from $S_n$ to $S_m$ with $n\geq m\geq 7$ in this article. I have some difficulty in understanding. First of all, why such condition is given $n\geq ...
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2answers
37 views

Suppose that $\cdot$ is associative and has an identity element. Show that an element $g \in G$ has at most one inverse

Let $(G,\cdot)$ be a group with $e$ its neutral element. For an element $g\in G$, there exists one inverse element in $G$, denoted by $g^{−1}$, such that $g\cdot g^{−1}=g^{−1}\cdot g=e$. Can this be ...
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2answers
25 views

Quotient Group question ambiguity about group order

$\mathbb{R}$ and $\mathbb{Z} $ are groups under addition. Show $a+\mathbb{Z} \in \mathbb{R/Z}$ is of finite order if and only if $a$ is rational. How can this be of finite order for any $a$? Let ...
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0answers
7 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
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0answers
38 views

Is there any survey paper or book for “Word Problem”? [on hold]

I found many many papers on this topic and reading them takes long time. I want to know for what kind of groups the word problem can be solved, and any other good general result. Is there any survey ...
3
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2answers
44 views

Show that $\langle x,y\mid xyx=yxy,x^3=y^2\rangle\cong\{e\}$.

Show that $\langle x,y\mid xyx=yxy,x^3=y^2\rangle\cong\{e\}$. So we are trying to show that $\langle x,y\mid xyx=yxy,x^3=y^2\rangle$ is isomorphic to the trivial group which contains only the ...
3
votes
2answers
32 views

If $G$ is a Finite Group such that $H\le K$ or $K\le H$ for all Subgroups $H,K$ of $G$, then $G$ is Cyclic and of order $p^n$ for some Prime $p$.

Since each subgroup $K$ is contained in some other subgroup $H$, we can list the subgroups of $G$ in ascending order $$\lbrace 1 \rbrace < G_1< G_2 < G_3\cdots< G_{k-1}<G$$ By ...
2
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1answer
48 views

“Every element of Sym$(n)$ has order at most $n$”

I was doing mini-test involving a True/False section and came across the following statement. Every element of $Sym(n)$ has order at most $n$ I admit I had gotten this incorrect as I had thought ...
2
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2answers
52 views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of ...
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0answers
39 views

Isomorphism between two groups

Is $S_5$ isomorphic to $A_5 \times \mathbb{Z}_2$? I tried to find an isomorphism between those 2 groups but i cannot so i think the answer is not.
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0answers
26 views

Is there a good way to break down the order of the centraliser in a symmetric group?

I recently rediscovered the rather nice formula for the order of the centraliser of a permutation in the symmetric group and its realtionship with conjugacy classes. I wondered whether we could say ...
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0answers
26 views

finding mistake in $\alpha^2(b)=b$

Let $X=\langle a,b|a^{2^m}=b^2=1,[a,b]=a^2\rangle , m\ge3$ and $\alpha \in Aut(X)$ (automorphis group of X) If \begin{cases} \alpha(a)=a^{2^{m-2}+1}b\\ \alpha(b)=a^{2^{m-1}}b \end{cases} and ...
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2answers
52 views

question asked around a weird concept

I am struggling with these questions. I dont know what is meant by the term system of representatives. anybody knows about these things?
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1answer
23 views

For $N, M \unlhd G$ relation between $MN/(M\cap N)$ and $N/(M\cap N)\times M/(M\cap N)$

Let $N, M \unlhd G$. Is $MN/(N\cap M)$ isomorphic to some subgroup of $$ N/(N\cap M) \times M/(N\cap M) $$ and how to prove this?
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0answers
19 views

Order's size (in bits) of an elliptic curve

I am trying to prove that, given an Elliptic Curve defined on $\mathbb{F}_p$ with $p$ a prime number, the order $q$ verifies: $|p| \le |q| \leq |p|+1$ where $|x|$ denotes the length in bits of $x$. ...
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0answers
13 views

Proving that rotation-inversion axis 2 and rotation axis n/2 induce rotation axis n.

How to prove that rotation axis of n/2 order and rotation-inversion axis of 2 order induce rotation-inversion axis of n order?
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votes
1answer
34 views

Can someone check my answers on group permutation and answer part (g) [on hold]

it would be great if someone could check my answers for Question 5 and answer part (g) Thanks you very much! Question 5: ...
0
votes
0answers
10 views

Proving that rotation-inversion axis n contains rotation axis n/2. [on hold]

How to prove that rotation-inversion axis of n order contains rotation axis of n/2 order when n is even?
0
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0answers
22 views

Kernel of group actions on a set $A$

Let a group $G$ act on set $A$. Show that if $b=g\cdot a$ for $a,b\in A$ and $g\in G$ then $G_b=gG_ag^{-1}$. Find the kernel of this action if G acts transitively on A. I dont know how to approach ...
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1answer
30 views

group theory dihedral group problem [on hold]

I am stuck in this problem. plz give some suggestion
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0answers
21 views

groups of colors in a colorful cube - combinatorics [on hold]

find natural number n, such that in every paint of a cube $$ 2^{[n]} $$ with the seven colors of the rainbow : a) there is 3 different groups $$ A, B , A \cap B $$ with the same color b) there is 3 ...
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0answers
62 views
+50

Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?

Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \in H\} \subset X;$$$X^H$ is the $H$-fixed point subspace of $X$. ...
7
votes
0answers
29 views

Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
1
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0answers
45 views

How to remember proofs in group theory [on hold]

I am new to algebra. I am facing a lot of problems in cramming all those proofs of groups. Can anybody help me to understand these proofs
0
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0answers
35 views

a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
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0answers
23 views

Soluble group of derived length 4

Is it true that a non abelian soluble group of derived length at most 4 is locally finite? It looks like this is the conclusion in an article, but I couldn't figure out why.
-1
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1answer
26 views

Abelian-by-(finite abelian) [on hold]

hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
3
votes
2answers
29 views

Why do “the Dynkin components of a weight play the role of eigenvalues with respect to the generators $H^i$ of the Cartan subalgebra”?

In the book "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists" by Jürgen Fuchs,Christoph Schweigert the authors write "In the description of representations, the ...
2
votes
1answer
32 views

Littlewood Richardson rules for the orthogonal group SO(d)

I have a question related to tensor products of Young diagrams. More precisely, I know the Littlewood Richardson rules for the general linear group GL(d) and would like to know the restriction of ...
0
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0answers
17 views

Automorphism of two members as Generator

Let $X=\langle a,b|a^{2^m}=b^{2^n}=1,[a,b]=a^{2^{m-1}}\rangle$, $m,n\ge 2$ If $\alpha \in Aut(X)$ (Automorphism Group of $X$) is defined as \begin{cases} \alpha(a)=a^{2^{m-1}+1}\\ ...
3
votes
1answer
27 views

Showing that a finite by cyclic group its automorphism group is finite

Let $G$ be a finite by cyclic group.Prove that its automorphism group $\operatorname{Aut}(G)$ is finite. A finite by cyclic group is a group G that has a normal subgroup $N$ such that $N$ is ...