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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Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
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Let $p$ be a prime, and $k \in \mathbb N$. What is the order of $U(p^{k})$?

Q: Let $p$ be a prime, and $k \in \mathbb N$. What is the order of $U(p^{k}) = |U(p^{k})|$? I understand that $|U(p)| = (p-1)$. With some experimenting I suspect that $|U(p^{k})| = (p-1)*p^{k-1} $. ...
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86 views

Power Automorphism of P Groups

The lower exponent p-central series for a p-group G is defined by $G=P_1(G) \geqslant P_2(G) \geqslant...\geqslant P_c(G)=I$, , where $ P_{i+1}(G)= [P_{i}(G) , G ] P_{i}(G)^{p}$ for $i \geqslant1$ and ...
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2answers
100 views

Textbook proof for uniqueness of inverse in a group

A Textbook of Abstract Algebra by Pinter gives the following proof of the property that an element in a group can have only one inverse (consider $*$ to be the operation and $e$ to be the identity ...
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1answer
648 views

Prove the order of an element divides the order of the group using cosets

I know how to use cyclic groups and Lagrange's theorem to prove this, but I don't know how to use the notion of cosets to prove this.
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1answer
250 views

What is the Betti number of a group?

I'm studying the Fundamental Theorem of finitely generated Abelian group, and it says that the number of factors equal to $\mathbb Z$ (textbook says it is the Betti number of the group) is unique up ...
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1answer
138 views

Second cohomology group of a perfect group

Consider a finite perfect group $G$ and a $G$-module, $U\left(1\right)$, on which $G$ acts trivially. Here $U\left(1\right)$ is the set of $1 \times 1$ unitary matrices over $\mathbb{C}$. Are there ...
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393 views

Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$

Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $Z_{2p}$ or $D_p$ . Gallian gives a proof as follows : They prove that G = $\langle a \rangle \...
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75 views

I have a question about groups of finite order.

I want to show that if a group G has finite even order then it must have an odd number of elements that are their own inverse, and if G has odd order then it has no elements of order two. I know this ...
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80 views

Given a set of generators of a group G, is there a method to find a presentation for G using those generators?

Suppose I have a group $G$, which I know is finitely presentable and infinite. (In particular, I have a presentation for it, though not the one I want). Suppose I have a small list of generators $\{...
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70 views

Orbits of action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$

I'm considering the action of $SL_m(\mathbb{Z})$ on $\mathbb{Z}^m$: if $A\in SL_m(\mathbb{Z})$ and $v\in\mathbb{Z}^m$, then $Av\in\mathbb{Z}^m$. My question is: what are the orbits of this action? I'...
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170 views

A diagram of group homomorphisms

Consider the commutative diagram of group homomorphisms: $$\require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @V{k}VV \\ C @>{g}>> D \end{CD}$$ If both $f$ and $g$ are injective ...
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1answer
44 views

How many groups are there of order 12?

All I could think of was the direct products of additive integer groups whereas $gcd(m,n) = 1$, such that $m$, $n$ elements of $\mathbb{Z}_{m} \times \mathbb{Z}_{n}$ . A hint would be really helpful.
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2answers
39 views

Defining a function and showing it is an automorphism

Let G be a group. Let $a \in G$ and keep it fixed. In terms of $a$, define a function $F_a$ from $G$ to itself by the rule $F_a(x) = axa^{-1}$. Prove that $F_a$ is an automorphism of $G$ I know it ...
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1answer
45 views

Solving a divisibility problem using group theory

Inspired by this I decided to show this: Let $P_n=\displaystyle\prod_{1\leq i<j\leq n} \big(x_i-x_j\big)$ where $x_1\,\dots\, x_n$ are arbitrary integers. Prove $n!\,\big|\, 2P_n$. I am ...
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1answer
130 views

Is the intersection of a core-free maximal subgroup with an abelian normal subgroup, trivial?

Let $G$ be a group, $H<G$ a core-free maximal subgroup and $A \triangleleft G$ an abelian normal subgroup. Question: Is it true that $A \cap H = \{e\}$
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1answer
48 views

Having trouble grasping the class equation as an explanation as to why a conjugate class's order divides the order of a group.

Suppose $|G|$ is a prime power $p^n$ and that $N$ is a normal subgroup of $G$. Show that $|y^G|$ is a power of $p$ whenever $y \in G$ Attempt: Firstly, I assume that $y^G = \{ gyg^{-1} | g \in G \}$'...
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1answer
60 views

Proof about coxeter groups

Prove: If a coxeter system $(W,S)$ is reducible, then it is the product of parabolic subgroups. Reducible system means the coxeter diagram is disconnected. Parabolic subgroup: let $S$ be the set ...
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1answer
71 views

Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook "...
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1answer
83 views

Direct Product Of Abelian Group

This is an assignment question: I just need know whether I am on the right track..... Let $G_1,…,G_n$ be groups. Prove that the direct product $G_1×⋯×G_n$ is abelian if, and only if, each of $G_1,…,...
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110 views

example of a subgroup of index $3$ which is not normal

Could you please give an example of a subgroup of index $3$ which is not normal ? I know every subgroup of index $2$ is normal but if index is $3$ , I have no idea whether all of the subgroups are ...
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69 views

Relationship between ord($a$) and ord($a^k$)

Let $a$ be an element of order $n$ in group $G$. Prove that if $m$ and $n$ are relatively prime, then $a^m$ has order $n$. (Hint: If $a^{mk}=e$, use the theorem: $a^t=e$ iff $t$ is a multiple of $n$, ...
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212 views

Let $G$ be an Abelian group with odd order. Show that $\varphi : G \to G$ such that $\varphi(x)=x^2$ is an automorphism

Let $G$ be an Abelian group with odd order. Show that $\varphi : G \to G$ such that $\varphi(x)=x^2$ is an automorphism. I was able to show that the $\varphi$ function is a homomorphism and one-to-...
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2k views

Counterexample for the normalizer being a normal subgroup

Let $G$ be a group, and $H$ a subgroup. $H$'s normalizer is defined: $N(H):=\{g\in G| gHg^{-1}=H \}$. Prove $N(H)$ is a normal subgroup of G, or give counterexample. Intuitively it seems to me ...
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What is explicit isomorphism between $H^2(G,\mathbb{Q}/\mathbb{Z})$ and $H_2(G,\mathbb{Z})$?

Let $G$ be a finite group. Then its Schur multiplier is the second cohomology group $H^2(G,\mathbb{Q}/\mathbb{Z})$, which is isomorphic to the second homology group $H_2(G,\mathbb{Z})$ (Proof can be ...
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79 views

An example of a topological space which is a group, but is not a “topological group”

Is there any example of a topological space $X$ which has a group structure, but the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are not continuous?
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76 views

Let $H$ be a subgroup of $G$ and let $a$ and $b$ belong to $G$. Then $aH=bH$ or $aH \bigcap bH =\emptyset$ [duplicate]

Let $H$ be a subgroup of $G$ and let $a$ and $b$ belong to $G$. Then : $aH=bH$ or $aH \bigcap bH =\emptyset$ ...... (1) Gallian gives the following proof which i have a little trouble understanding :...
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42 views

Property of isomorphic normal subgroups

Suppose $A$ and $B$ are two normal subgroups of given $G$ group and $A \cong B$. I'm wondering how to prove that $\phi(A)\cong B$ where $\phi \in \operatorname{Aut}(G)$.
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1answer
62 views

Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. $U\...
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593 views

Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.

Let G be a group, where $(ab)^3=(a^3)(b^3)$ and $(ab)^5=(a^5)(b^5)$. Prove that $G$ is abelian group. Thank you in advance. Any help is appreciated.
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Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.

Let $A$ be a group, where $a^2=1$ and $a$ belongs to $A$. Prove that this group is commutative. Thank you for help.
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1answer
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Find the center of the group GL(n,R) of invertible nxn matrices. [duplicate]

Find the center of the group GL(n,R) of invertible nxn matrices. Please can someone please help me? I know that by definition the center Z of a group is defined by Z(G) = { ag = ga | for all a in G}. ...
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Closure of divisibility in denumerator, under sum of fractions

I have to prove that for a fixed positive integer n, the subset A of Q consisting of rationals with denumerator that divide n under addition, forms a group under addition. I just did that it's ...
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1answer
609 views

A solvable group with composition series is finite

I want to prove the following. A solvable group G with composition series is finite From the definition of a solvable group I know there exists a normal series: $\lbrace 1 \rbrace \le G_{n} \leq G_{...
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Unicity inner automorphism symmetric group

I need to prove that the center of the symmetric group $S_n$ with $n\geq 3$ is trivial. I chose to use Lagrange's theorem: bearing in mind that the center is a subgroup, it is easily found that $\#Z(...
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1answer
85 views

Show that the orbit of $H$ containing $x$ is equal to the right coset $Hx$

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $X$ be the set of elements of $G$. Let $ \ast : H \times X \to X$ be given by $$ h \ast x = hx (h \in H, x \in X)$$. QUESTION: Let $x \in X$. ...
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1answer
288 views

Show that matrix under addition is isomorphic with the group of complex numbers under addition

Q: Let $M = \{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} | a, b \in \mathbb R \}$. Show that $(M,+)$ and $(\mathbb C,+)$ are isomorphic. Show that $(M^{*},*)$ and $(\mathbb C^{*},*)$ are ...
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2answers
102 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
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Quotient groups and normal subgroups

I am wondering if there is some characterization of the normal subgroups of a quotient group. More precisely let $G$ be a group and $H$ a normal subgroup. Let $U$ be a normal subgroup of the quotient ...
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1answer
44 views

Prove $o(g) | |G|$ using Lagrange's Theorem

I'm confused of the following steps: How did the lecturer form the subgroup $H$? How do we know that $H$ is cyclic? if anyone could please explain, thank you
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1answer
47 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
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Raising multiplied group elements to a power

If for example we have $(aba^{-1})^n$, how do you go about expanding this to show that it's the same as $a^nba^{-n}=ab^na^{-1}$?
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34 views

Given $H \le G$ and $P \in \text{Syl}_{p}(G)$, to show that $\exists a \in G, aPa^{-1} \cap H \in \text{Syl}_{p}(H)$.

The problem is as follows: $G$ is a finite group. $H \le G$ and $P$ is a Sylow $p$-subgroup of $G$. To prove that: $\exists a \in G$ such that $aPa^{-1} \cap H$ is a Sylow $p$-subgroup of $H$. ...
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86 views

Product of cyclic groups

How can you quickly tell that the product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_3$ has a 2-subgroup containing an element of order 4? Also, I don't understand the notion of multiplying ...
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1answer
807 views

If in a group $G$, we have $a^5 = e$ and $aba^{-1} = b^2$ for some $a$, $b$ in $G$, then what is the order of $b$?

If in a group $G$, we have $a^5 = e$ and $aba^{-1} = b^2$ for some $a$, $b$ in $G$, then what is the order of $b$?. Here $e$ denotes the identity element in $G$.
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Why $a^p\in N$ if $[a]^p=[e]$ holds in $G/N$

Suppose there exists an element in $G/N$ of order $p$, where $p$ is a prime number. In other words, there exists an $a\in G$ such that $[a]^p=[e]$, where $[a]\neq[e]$. Then why $a^p$ belongs to $N$? ...
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1answer
57 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
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40 views

(possibly duplicated question) Prove the Cauchy theorem using mathematical induction

Here are some of my questions: 1) If the order of G is one, then there is no such p as stated in the above article, so why say that the theorem is true here? 2) where exactly is the starting point? It ...
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1answer
23 views

How can we characterize the left and right cosets of this subgroup?

Let, for real numbers $a$ and $b$, the mapping $f_{a,b} \ \colon \mathbf{R} \to \mathbf{R}$ be defined by $f_{a,b}(x) \colon = ax+b$ for all $x \in \mathbf{R}$. Let $G \colon = \{ f_{a,b} \ | \ a, ...
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1answer
62 views

What about the index of this subgroup? [duplicate]

Let $G$ be a group, and let $H$ be a subgroup of finite index in $G$, and let $N \colon = \cap_{x \in G} \ xHx^{-1}$. Then $N$ is clearly a subgroup of $G$ which is contained in $H$ and such that $aNa^...