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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Intersection of two normal subgroups of a group

Let G be a group, and let A,B be normal subgroups of G. If $a \in A$ and $b \in B$, does this mean that $ab \in A \cap B$?
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3answers
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Textbooks for Groups & Rings [closed]

Please I need suggestions on the best textbooks to help me comprehend this Groups and Rings and relate it with the rudimentary aspect of Set Theory
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6answers
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Order of elements in abelian groups

How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$? ...
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Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
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37 views

Permutation representations of symmetric group of small order

Thanks for any comments or help. What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
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2answers
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Prove that $H\cap C$ is non-empty for every conjugacy class of $G$

Let $G$ be a finite group and $\phi\in Aut(G)$ such that $\phi^q=id$ where $q$ is prime and does not divide $|G|$. Moreover $\phi$ preserves conjugacy classes of $G$. Then consider $H=\{g\in G : ...
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2answers
33 views

Is there a difference between modulo groups with and without asterisks ($\mathbb{Z}_{38}$ vs $\mathbb{Z}_{38}^*$)?

I know modulo group $\mathbb{Z}_{38}$ but I saw it with a star in some question: $\mathbb{Z}_{38}^*$. Is it is the same as $\mathbb{Z}_{38}$ or a different group? If it refers to the same group does ...
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1answer
25 views

Need information on the following multi-group homomorphic structure

For the sake of simplicity I will describe the problem with a three group structure. Suppose there are three groups $G_1$, $G_2$ and $G_3$. Suppose also there is a binary map $M:G_1\times G_2\to G_3$ ...
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1answer
20 views

Help showing $H_n=\sum_{h \in H}h \in F[H]$ is a simple submodule

Given $F$ a field and $H$ a finite group and $H_n=\sum_{h \in H}h \in F[H]$, I'm trying to show that the left ideal $(H_n)\subset F[H]$ is a simple submodule. Now I'm not really sure what the ideal ...
3
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1answer
55 views

Why can quotient groups only be defined for subgroups?

I know that for the operation on cosets to be well-defined one requires normality. But why is it a requirement (with $G / N$) that $N$ be a subgroup or even a subset of $G$? Surely all that is ...
3
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What dihedral subgroups occur in the affine general linear group $AGL(2,3)$

I am interested in the subgroup structure of the affine general linear group $AGL(2,3)$, in particular I want to know if they could have dihedral subgroups other then $D_3$ and $D_4$, i.e. the ones of ...
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0answers
25 views

$S_n$ is isomorphic the permutations of the identity matrix?

Prove that the set of permutations $S_n$ is isomorphic to the group of invertible square matrices of order $n$ where each row has $n-1$ zeros and $1$ in one place. This is very intuitive to me, ...
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1answer
42 views
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41 views

Index of free group of rank $k$ in free group of rank $n$

I thought about the following question What is the index of the free group of rank $k$, denoted $F_k$, in the free group of rank $n$, denoted $F_n$? Let's say for the moment, $k < n$, and ...
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Exercice about the group of unit $(\mathbb Z/21\mathbb Z)^\times $.

Let $(\mathbb Z/21\mathbb Z)^\times$ the group of units of $\mathbb Z/21\mathbb Z$. 1) How many element in $(\mathbb Z/21\mathbb Z)^\times$ ? 2) Is it isomorphic to an abelian group ? 3) Is it ...
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1answer
39 views

Finding the homomorphisms $S_3 \to \Bbb Z / 6 \Bbb Z$

I have to find explicitly (i.e. as operated on the element of the domain) the homomorphisms (of groups) from the symmetric group $S_3$ to $\Bbb Z / 6 \Bbb Z$. Do I study the possible kernels of the ...
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1answer
54 views

Index is multiplicative

Let $G$ be a group and satisfies minimal condition on subnormal subgroups. Further let $H, K\trianglelefteq G$, such that $H\leq K$ and $[G:H]$ is finte, then can we say something about $[G:K]$ ?. ...
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238 views

A theorem from the theory of groups

Let $K$ be a (not necessarily normal) subgroup of the group $G$ : $K < G$ A fixed element $g\in G$ can act, from the left, on all elements of $G$, thus generating a bijection of $\,G\,$ onto ...
4
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2answers
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Conjugate elements of $SO(3)$ group

Composition of two rotations in 3d space yields another rotation $$R_1 R_2 = R_3, $$ and I can understand this by help of some figures in my book. So, the rotations in 3d space forms group. Then ...
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1answer
27 views

A description of the transporter $\operatorname{Tran}_G(H,K)$ for subgroups $H\le K$

Let $H$ and $K$ be subgroups of a group $G$. The transporter of $H$ into $K$ is the set of all $g\in G$ that conjugate $H$ into $K$: $$\operatorname{Tran}_G(H,K)=\{g\in G\mid gHg^{-1}\le K\}$$ ...
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1answer
38 views

Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...
5
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3answers
473 views

Symmetries of a graph

Determine the number of symmetries in the following graph: What are the general things you should do when finding such symmetries? Usually I would label all of them from $1...n$ and note their ...
5
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2answers
156 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
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1answer
53 views

Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
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2answers
30 views

Inverse of the element in the multiplicative group

I need to show that if $k\in \Bbb Z^*_n$ has an inverse with respect to the multiplication modulo, $n$, then $k,n$ are coprime. Can anyone give me a hint how to use the fact that $k$ has an inverse ...
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1answer
28 views

If $\alpha,\beta \in S_{n}$, and $\alpha\beta = \beta\alpha$, then $\beta$ permutes those elements left fixed by $\alpha$.

Here is my solution. Let ${a_1,...,a_k}$ represent all the integers that are permuted by $\alpha$, and let ${a_{k+1},...,a_{k+j}}$, where $k+j \leq n$, be all the elements that are left fixed by ...
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1answer
24 views

Representatons of dimension $1$ on $D_4$

Prove that there are $4$ distinct representations on $D_4$ with dimension $1$ (where the field is $\mathbb{C}$). We have just started learning representations. Getting this question, what ...
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34 views

a representation of permutation groups

The smallest degree of faithful permutation representations of $S_k$ for $k\geq 6$ have degrees $k$ (natural action), $2k$ (imprimitive action on cosets of $A_{k−1} )$, and $k(k − 1)/2$ (action on ...
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1answer
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Can the possible groups by determined by hand?

Suppose, $G$ is a group of order $12$ containing an element $a$ with order $4$. Can I show the following facts by hand ? The group is either cyclic or isomorphic to the group $C3:C4$ $a^2$ is the ...
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2answers
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Is there a cyclic group such that is isomorphic to Z∗16? [closed]

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?
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0answers
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Number of non isomorphic groups of order 10

Find the number of non-isomorphic groups of order 10. I tried using fact that since there are no non abelian groups of order 10, so we are left with abelian groups which will be $\Bbb Z_2 ...
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1answer
25 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...
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An inequality for the minimal number of generators of a finite group [migrated]

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
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1answer
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Factorization of conjugacy equation's solutions in Monoids

01-28 Update: In the first version I was claiming that the authors were not explicitly or implicitly but I was wrong so I change my question [long explaination at the end of the question] Two ...
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2answers
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$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
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Problem 4 of Section G of Chapter 13 of Pinter's Book of Abstract Algebra

First, some background info/context: Let $G$ be any group of order $10$. Then, by Cauchy's Theorem, there are elements $a, b \in G$ such that $\text{ord}(a)=2$ and $\text{ord}(b)=5$. Since ...
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1answer
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Looking for some insight or explanation in Normal group and similar matrix

I need someone can give me some insight in Normal group and Matrix Similarity. Normal subgroup $g \in G \text{ and } N \lt G$ $gNg^{-1} = N$ Matrix Similarity $B = PAP^{-1}$ It seems to me ...
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1answer
44 views

Prove that (possibly) infinite group of all invertible maps of X to itself is not Abelian.

I have this question on my assignment and I this fact seems trivial to me, but I can not come up with a rigorous proof. I thought to go by contradiction: Assume such a group $G$ is Abelian -> ...
4
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2answers
81 views

Why isn't a set like $\{0,1,3,6,8\}$ a subgroup of $\mathbb{Z}_9$?

Why are there only 3 subgroups of $\mathbb{Z}_9$? What about $\{0,1,3,6,8\}$? There is an identity and inverse for each element in that subset.
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0answers
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Group Action: Group $(\mathbb{Z}, +)$ acting on $\mathbb{R}$

In my group theory notes I have the following: The Group $(\mathbb{Z}, +)$ acts on $\mathbb{R}$ as follows: $m\in \mathbb{Z}$ and $r\in\mathbb{R}$: $m.x \to (-1)^mr$ in this notation $m.x$ ...
4
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4answers
135 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
4
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1answer
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Group theory commutator and solvable groups

let G be a group such that it contains 2 members $a, b \in G$ that statisfy: $a = p^{-1} b p$ where $p \in G$ $a = q^{-1} [a,b]q $ where $q \in G$ $a,b,[a,b]\neq e$ where $[a,b]$ is the commutator ...
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Problems in Wikipedia about symmetry group of the non-equilateral isosceles triangle.

I think this wikipedia article - https://en.wikipedia.org/wiki/Symmetry_group#Two_dimensions - is wrong when it states that "$D_2$, which is isomorphic to the Klein four-group, is the symmetry group ...
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Can the number of order-sequences of a group of order $n$ be determined efficiently?

Let $n\ge 1$ and $G$ a group of order $n$. Determine the order of every element of $G$ and list the orders increasing. Can the number of possible order-sequences be determined efficiently ? For ...
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1answer
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Proving isomorphism [duplicate]

I want to prove at any group of order $4$ is isomorphic to either $\Bbb Z_4$ or $\Bbb Z^*_8$. I know that these two groups are not isomorphic to each other because they have different order, but I ...
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2answers
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When $n$ is odd, $\langle (123),(12…n)\rangle$ generates $A_{n}$

Working with the knowledge that the set of 3-cycles generates $A_{n}$, the basic idea is to express any 3-cycle as a word in $(123)$ and $(12...n)$ when $n$ is odd. Not knowing how to progress, I ...
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1answer
36 views

representation of symmetric group

Please help me for the following statement. I do not understand what does it mean centralizer of a representation. The imprimitive transitive representation of symmetric group $S_n$ of degree 2n has ...
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1answer
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Determining which elements of a matrix group form a one-parameter subgroup

We have just learned about one-parameter subgroups in my Algebra class and I am not sure if I am approaching the following proof in the right way. Problem Statement: Let $G$ be a group of real ...
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Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. [duplicate]

Let $p$ be a prime. Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. I know that the matrix $2\times 2$ is given by this formula ...
0
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1answer
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Abelianization of $\mathbb{Z}_2*\mathbb{Z}_3$

Intuitively it has to be $$\text{Ab}(\mathbb{Z}_2*\mathbb{Z}_3)=\mathbb{Z}_2\times\mathbb{Z}_3$$ here is my approach on how to prove it $$\mathbb{Z}_2=P(a\mid a^2),\mathbb{Z}_3=P(b\mid b^3)\Rightarrow ...