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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problem about inner semidirect product

Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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0answers
40 views

Maximum order of element in group of units in a ring

Let $s$ be a natural number and $U(s)$ be a group of units in the ring $\mathbb{Z}/s\mathbb{Z}$. Let $\phi(s) = 2^{k_1}p^{k_2}$, where $p$ is a an odd prime number. I don't understand why the maximum ...
4
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3answers
22 views

To show that Z(G) = $\cap_{a \in G} C(a)$

To show that Z(G) = $\cap_{a \in G} C(a)$ (Intersection of all subgroups of form C(a)) Let $a \in Z(G)$. Then $ax=xa$ for all $x$ in G. IN particular we can say that $ax_1=x_1a$ and $ax_2=x_2a$ and ...
4
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0answers
32 views

Minimal Normal Subgroups of an elementary abelian p-group

Can you explain/prove, why the number of minimal normal subgroups of an elementary abelian $p$-group of order $p^n$ (for instance of $\mathbb{Z_p}^n$), is exactly $(p^n-1)/(p-1)$? I know that it seems ...
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1answer
26 views

is the following equality about group true

Let $G,G_1,G_2,H$ be groups. Is the following equality true: If $G=G_1\cap G_2$ then : $ |H/G|= |(H/G_1)\times (H/G_2)|$ A specific example is: Is $|\mathbb{Z}/mn\mathbb{Z}| = |(\mathbb{Z}/m ...
2
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1answer
34 views

When will the classification of a property of a group be complete

I am studying pronormal Hall $\pi$-subgroups of a finite group $G$ All Hall $\pi$-subgroups of a finite solvable groups are known to be pronormal as this follows from Hall's theorem Recently, it has ...
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0answers
23 views

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$ [duplicate]

Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even. Same question have been asked ...
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0answers
35 views

Show that in a group order of element is less than or equal to order of group [closed]

Show that in a group order of element is less than or equal to order of group. This is a Question from Gallian. Please provide hints on how to start this? THanks
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0answers
31 views

How to proof that if for every proper subgroup $H$ of $G$, $N(H)\neq H$ then $G$ is nilpotent group? [closed]

Let $H$ be a proper subgroup of group $G$. If for every $H\leq G$, $$N_G(H)\neq H$$ then prove that the group $G$ is nilpotent.
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0answers
29 views

Why is a nontrivial finite group is nilpotent if every maximal subgroup is normal?

In the following proof I understand the above proof up to the part where it says "by Sylow theory $N(M)=M$", could someone explain to me why is this true. We have just started learning about group ...
2
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2answers
57 views

Prove that if $G$ is cyclic and infinite then $G$ is isomorphic to $\mathbb{Z}$

Assume $G$ is generated by $a$, so $G = \langle a\rangle$. Since $G$ is infinite for all $m \in \mathbb{Z}$, $a^m \neq e$. Suppose $a^h = a^k$ then $a^h\cdot a^{-k} = a^{h-k} = e$, but this is a ...
1
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1answer
43 views

Chain of subgroups

Given a chain of subgroups $K<H<G$ a.) Prove or disprove: if $K$ is a normal subgroup of $H$ and $H$ is a normal subgroup of $G$, then $K$ is a normal subgroup of $G$. b.) Prove or disprove: ...
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0answers
50 views

Are all subgroups of the fundamental group of a compact smooth manifold finitely generated?

And if not, is there a way to assign a size to a subgroup by considering the compactness of the corresponding covering space?
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3answers
94 views

Is cyclic $\mathbb{Z}_4 \times \mathbb{Z}_ {12} \times \mathbb{Z}_9$?

I find on internet this: $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic if and only if $\gcd(m,n)=1$. Then I do the next steps: gmc(4,12,9) is 1. Then I assume that $\mathbb{Z}_4 \times ...
3
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2answers
54 views

Show that open interval $(-1,1)$ is isomorphic to $(\mathbb{R},+)$

Define group structure on $G=(-1,1)$ by $$a*b=\frac{a+b}{1+ab}$$ for any $a,b\in G$. Show that $G$ is isomorphic to $\mathbb{R}$ under addition. I've tried the obvious maps $f:G\rightarrow ...
2
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0answers
26 views

subgroup of a semidirect product

I'm really lost with this problem and I really need your help: Let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$, and let $H\leq G$ with finite index in G. I have to prove that there is a subgroup $U$ of ...
2
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1answer
26 views

quotient groups and SLOCC

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
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0answers
33 views

Simple subgroups of a symmetric group

In class we used in an exercise that "the only simple subgroup of a symmetric group (if it has one) is the alternating subgroup". But I don't understand where this comes from. Can someone help me?
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2answers
25 views

Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in ...
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1answer
38 views

If a finite group $G$ has a normal subgroup $N$ of order $p$ where $p$ is the smallest prime divisor of the group order, then $Z(G)$ is nontrivial

Let $G$ be a finite group and $p$ the smallest prime number with $p \mid |G|$. I want to show: if $G$ has a normal subgroup $N \trianglelefteq G$ of order $p$, then $G$ has a nontrivial center. My ...
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2answers
40 views

Finding the Left and Right Cosets in $A_4$

I'm really struggling with a Group theory class and would love some help. HW Question is as follows. Consider the subgroups $H = \left<(123)\right>$ and $K=\left<(12),(34)\right>$ of ...
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1answer
47 views

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$?

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$? (With lexicographical order) Thanks
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1answer
36 views

Galois field of order 2 constituting a Boolean algebra

We know that the the set $\{0,1\}$ constitutes a Boolean Algebra over the usual $OR$ and $AND$ operations. However, because of the lack of an additive inverse for $1$ this does not produce a Galois ...
4
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1answer
61 views

Is an abelian subgroup of a finitely generated group finitely generated?

Let $H <G $ be groups, G finitely generated and H abelian. Is H then finitely generated?
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2answers
418 views

Does an abelian subgroup inject into the abelianisation of the whole group? [closed]

If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?
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1answer
27 views

Doubt in Dihedral group $D_4$ regarding reflections [duplicate]

This question is from Gallian Page 69 Q 9. Suppose a subgroup od D_4 contains H and D. I want to show that these two generated whole of $D_4$. Now rotation will generate other rotations which is ...
3
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1answer
52 views

If a is a group element and a has infinite order, prove that $a^m\neq a^n$ when $m \neq n$

If $a$ is a group element and $a$ has infinite order, prove that $a^m\neq a^n$ when $m \neq n$. (Gallian, Contemporary Abstract Algebra, Exercise 19, Chapter 3.) To prove that $m \neq n$ implies ...
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42 views

In a group we have $a^6=e$. What are possibilities for order of a? [duplicate]

In a group we have $a^6=e$. What are possibilities for order of a? I think order of a can be 2 or 3. Because if order of a is 2. Then $a^2=e$. Multiplying by $a^4$ we get $a^6=e$ which is true. ...
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0answers
33 views

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, is it true that $ G A_n=S_n$ [closed]

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, then is it always true that $ G A_n=S_n$.
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0answers
28 views

Self conjugation on a group with $2p$ elements

Let $G$ be a group with $2p$ elements where $p$ is an odd prime. Also let $Z(G)=e$, where $e$ is the identity element. Prove that there is a conjugationclass with $p$ elements. My attempt: Because ...
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1answer
30 views

Name of a family of Coxeter groups

From the following image I know that the first of group is the symmetric group of rank $n$ and the second is known as the Hyperoctahedral group. I want to know if someone knows the name of the ...
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0answers
24 views

general spin groups as quotient of spin group

I am trying to understand Spin groups, but unfortunately I am now stuck on something I came across! I hope someone could clarify or maybe point me to some text where I can read about it! I searched in ...
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0answers
17 views

Hall $\pi$-subgroups of a normalizer

Let $G$ be a group with $A\unlhd G$ and $H\in$ Hall$_\pi$(G) Consider $N_G(H \cap A)$. Then $H$ is a Hall $\pi$-subgroup of $N_G(H \cap A)$ Let $K$ be another Hall $\pi$-subgroup of $N_G(H \cap A)$ ...
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1answer
79 views

Counting homomorphisms of groups

How many homomorphisms are there from $A_{5} × S_{3} × A_{4}$ to $\mathbb Z/2\mathbb Z$? I know that $Z/2Z$ is generated by ome element and of order two. I know the respective order of the rest ...
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2answers
77 views

Is $\mathbb{Z}$ contained in $\mathbb{Z}_p$? [closed]

I was wondering if $\mathbb{Z}$ is contained in $\mathbb{Z}_p$, the group of integers modulo $p$? As I can take every integer and send it to it equivalence class I believe that I could be possible?
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1answer
28 views

Isomorphism between finite groups with specific property is unique

I am looking for a clever justification for a statement I believe to be true. I would like to show that given an isomorphism, $\Phi$, between two finite groups, if it is known that $\Phi(a) = b$, then ...
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1answer
19 views

$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]|$ is even

Let $G$ be a finite group. My problem is: $$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]| \mbox{ is even}.$$ I know that $\hom(G, \Bbb Z/2)=\hom(G^{ab}, \Bbb Z/2)$, where $G^{ab}:=G/[G,G]$. If ...
0
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2answers
56 views

Prove $(G,.)$ group.

If $$G = \left\{ \begin{pmatrix} a & b \\ b & a \end{pmatrix} a,b\in \mathbb{R} : a,b \neq 0 \right\} $$ Prove that $(G,.)$ group. I proved binary operation and associative , but what ...
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0answers
16 views

Center of Applications from X to a group G

I have to demonstrate: Let B a group and X a nonempty set. So, $$Z[Aplc(X,G)]=Aplc[X,Z(G)]$$ where $Z$ is the center of the group. And $Aplc(X,G)$ is the set of all the applications from X to G. ...
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2answers
69 views

Is empty set $\emptyset$ group?

I want to ask about empty set $\emptyset $ is group or not ? I know any set G be group if : For every $a,b,c\in G$ 1- $"*"$ binary operation. $a*b\in G$ 2- $"*"$ associative. $(a*b)*c=a*(b*c)$ ...
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1answer
42 views

Number of Possible Configurations

I have an embarrassingly simple problem that I'm not confident that I'm answering correctly. Say you have a 3 by 3 grid, where any number of spaces in the grid can be colored in, including all or ...
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0answers
16 views

Hall $\pi$-subgroup property

Let $A$ be a normal subgroup of $G$ and $H$ be a Hall $\pi$-subgroup of $G$ Let $K$ be a Hall $\pi$-subgroup of $N_G(HA)$. I need to show that $K\leq HA$ and $KA = HA$ Since $[G : H] = [G : ...
2
votes
1answer
39 views

Character theory - exercise 5.16 from Isaacs

Hi I am trying to solve the following exercise. Let $H$ be maximal subgroup of a finite group $G$ and let $\chi=(1_H)^G$. Let $\psi$ be a non-principal irreducible constituent of $\chi$. Then $Ker ...
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0answers
24 views

Character theory - exercise 3.4 from Isaacs

Let $G$ be a simple group and suppose $\chi\in Irr(G)$ with $\chi(1)=p$ a prime, Show that a Sylow $p-$subgroup of $G$ has order $p$. The indication provided is: if the Sylow $p-$subgroup $P$ is ...
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60 views

How to define a function when proving two groups are not isomorphic?

So I am given two groups and I need to prove that they are not isomorphic. According to the definition of isomorphism, the condition $f(xy) = f(x)f(y)$ should be satisfied. Does it really matter how ...
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40 views

A request of a journal theorem

I am reading the paper Groups whose proper subgroups are finite-by-nilpotent by Maoqian Xu and he has some references that I can't find. Does somebody have (B. Bruno, On p-groups with ...
2
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1answer
31 views

Subgroup of prime index

Let $G$ be a solvable group and $p$ is a prime such that $p\mid |G|$. Does there exist a subgroup $H$ of $G$ such that $[G:H]=p$ ?
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21 views

Find character of exterior power.

Consider the symmetric group $S_4$ and let $\phi: S_4 \to \text{Aut}(V)$ be a representation of $S_4$ of degree $3$ whose character $\chi$ is given by $$1^4 \mapsto 3$$$$2^2 \mapsto -1$$$$(3,1) ...
2
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1answer
38 views

Prove that $\alpha$ is an automorphism of $Z_n$.

Let $ r \in U(n)$[We define $U(n)$ to be the set of all positive integers less than $n$ and relatively prime to n]. Prove that the mapping $ \alpha : Z_n \rightarrow Z_n $ defined by $ \alpha(s)=sr$ ...
1
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1answer
16 views

How does a symmetric group act on a set?

Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq 3 \}$ by $\sigma \cdot (i,j) = (\sigma(i), \sigma(j))$. Find the orbits of $S_3$ on $A$. So I know I have to find a set $\{\sigma \cdot ...