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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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
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1answer
9 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
19 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 $\lambda$ ...
0
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1answer
18 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
30 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
29 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
23 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
37 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
41 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
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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
45 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
49 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 ...
-1
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0answers
38 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
24 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
25 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
18 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
11 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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1answer
33 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
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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?
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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
29 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
46 views

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 $X$. ...
6
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0answers
27 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 ...
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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
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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
22 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
votes
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
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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
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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 ...
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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
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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 ...
0
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1answer
35 views

Intersection of subgroup of finite index with infinite subgroup is infinite [on hold]

Let $H\subseteq G$ be a subgroup of $G$ of finite index. Further, $F\subseteq G$ is an infinite subgroup of $G$. Prove or disprove: $H\cap F$ is infinite.
1
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1answer
43 views

If $|xH|$ has order $n$, then there is an element $y$ with $|y|=n$ and $xH=yH$

Let $G$ be a group, and let $H$ be a normal subgroup with $|H|=m$. Suppose $n$ and $m$ are relatively prime. If $|xH|$ has order $n$, we wish to find an element $y$ with $|y|=n$ and $xH=yH$. It is ...
2
votes
3answers
104 views

Is there any good software that solves equations of permutation group elements?

I need to solve equations of permutation group elements (elements of $S_n$) that may not may not have solutions. The number of equations generally exceeds the number of variables. Is there any good ...
2
votes
1answer
24 views

Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
0
votes
1answer
16 views

Is there any way I can show $\theta(x)\theta^2(x)=x^{p-1}$, for $\theta\in \operatorname{Aut}(C_p)$ having order $3$

So I am not sure if I am being dim but I am stuck on what seems like a straightforward group theory argument. Suppose we have some $\theta\in \operatorname{Aut}(C_p)\cong C_{p-1}$ such that ...
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0answers
31 views

Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
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3answers
36 views

Axioms for Group Action

Reading the Wikipedia article on group action, I am wondering, why are the axioms stipulating that a group action obey both "compatibility" and "identity"? If a group action is merely a group ...
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0answers
28 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
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0answers
26 views

For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?

Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ? ( One problem I'm ...
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0answers
11 views

Complexification of a Lie algebra representation in terms of weights?

EDIT: I found in this book the sentence: The weight system of a real representation of $G$ is defined to be the weight system of its complexification I think if someone can explain what this ...
0
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1answer
47 views

Determine number of elements of order 12 of a group

Let's say we have a commutative group G that's specified by generators and relations. We find that the group G normal form is: $Z_2\times Z_6\times Z_{12}$ and that the elementary form is $Z_2\times ...
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36 views

Where to the degrees of freeedom go when a complex representation becomes a real representation of a subalgebra?

As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional ...
0
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1answer
23 views

Meaning of $^sB$, s an element, B a subgroup

Let $G = SL_2(\mathbb{F}_q)$, $B$ the subgroup of all upper triangular matrices, $s = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. What does $^sB$ mean? I read it from page 4 of C. ...
0
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2answers
63 views

If $\phi$ is an isomorphism, $\phi(g)^n = 1 \iff g^n = 1$. Doesn't this hold for homomorphisms too?

I need to prove that for an isomorphism $\phi$, the following is true: $$\phi(g)^n = 1 \iff g^n = 1.$$ We know that $$g^n = 1 \implies g\cdot g \cdots g = 1\implies \phi(g\cdot g \cdots g) = ...
9
votes
1answer
55 views

An element of $GL_n(\mathbb F_p)$ cannot have order $p^2$ if $n < p$

I'm preparing for my graduate program's entrance exams, and I came across this problem when studying. Our study group came up with a solution, but I wanted to ask if it was actually correct, since ...