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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Nonabelian $2$-groups whose derived length $l > 2$

Can we classify all nonabelian $2$-groups whose derived length is at least $3$?
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A good book for beginning Group theory

I am new to the field of Abstract Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and ...
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2answers
30 views

Isomorphism between $G$ and $\mathbb{Q}^{*}$

Let $\{G_{n}\}_{n\in \mathbb{N}}$ be a family of additive groups with $G_{1}=\mathbb{Z}_{2}$ and $G_{n}=\mathbb{Z}$ for $n\geq 2$ $$G=\bigoplus_{n\in \mathbb{N}}G_{n}$$ I want to prove that $G\cong ...
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$G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate

Let $G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate and any proper subgroup is contained in a maximal subgroup . Then is $G$ cyclic ? I ...
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12 views

Does irreducibility of a representation imply irreduciblity of all restricted representations?

Let $G$ be a group with a subgroup $H$. Then any representation of $G$ can be restricted to $H$. If the $G$ representation is irreducible then should the $H$ representation also be irreducible? If ...
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Are all countable torsion-free abelian groups without elements of infinite height free?

The height of an element $g$ in an abelian group $G$ is the largest $n\in \mathbb{N}$ such that there exist an element $h\in G$ such that $n*h=g$. If $g$ has no such largest integer than $g$ is of ...
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14 views

What's the motivation of the third condition of Nielsen-reduced?

The third condition of Nielsen-reduced is: $v_1v_2\ne$l and $v_2v_3\ne$l implies |$v_1v_2v_3$|> |$v_1$|-|$v_2$|+|$v_3$|.But I can't see why it's defined by this, how did he come up with such an idea?
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1answer
21 views

perfect groups with non-trivial group homology over rational coefficients

A group $G$ is perfect if $G=[G,G]$. For perfect groups, we know that the first group homology $H_1(G, \mathbb{Z})=G/[G,G]=0$. A group $G$ is called acyclic if its group homology $H_i(G, ...
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1answer
20 views

Commutator subgroup $[H,K]$, $H, K$ subgroups of a group

How could I show that $[H,K]$ is a normal subgroup of $\langle H, K \rangle$? Also that if $H$ is generated by $X$ and $K$ is generated by $Y$, then $[H,K]=\langle g[x,y]g^{-1} | x \in X, y\in Y, g\in ...
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For the special class of $T$-subgroups in a certain quotient group the normal subgroups intersect certain subgroup

Let $G$ be a finite group and $U \le G$ be a subgroup of odd order which has index two in its normalizer and $U^g \ne U$ implies $U^g \cap U = 1$. Write $N_G(U) = TU$ with $T = \langle t \rangle$ for ...
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1answer
41 views

To prove that $G$ is the group the condition is not necessary $\forall a,b, c \in G(ba=ca\to b=c)$.?

$1.$ Let $G$ be a finite semigroup such that $\forall a,b, c \in G(ab=ac\to b=c)$. Then $G$ is Group. ? I know the following result : If $G$ be a finite semigroup such that $\forall a,b, c \in ...
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0answers
33 views

Classify all finite groups with property [on hold]

Classify all finite groups $G$ with the following property: for every $H\vartriangleleft G$ there exists $K<G$ such that $G/H$ is isomorphic to $K$. My poor abstract-algebraic imagination doesn't ...
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35 views

A relation between a group and its subgroups [duplicate]

Let be $H$ a proper subgroup of finite group $G$. Who can we show that $G\not=\cup_{a \in G}aHa^{-1}$?
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2answers
19 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
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1answer
32 views

When does normal maximal subgroup have prime index?

Given finite group $G$, a normal maximal subgroup $H$, when is $[G:H]$ a prime? If $G$ is nilpotent, then the statement is true. But I am not sure about other $G$. Is there any counter-example ...
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24 views

Sylow counting to show group is isomorphic to semidirect product

Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$. Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and ...
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32 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
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2answers
22 views

Order of group $G = \{A\in M_2(\mathbb{Z}_p): \mathrm{det}A= \pm 1 \}$

Also, $p>2$ is a prime number. Firstly, it's obvious that $G \leq GL_2(\mathbb{Z}_p)$, and we know that $|GL_2(\mathbb{Z}_p)|=p(p^2-1)(p-1)$. Next, we define the homomorphic map ...
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26 views

Using a Sylow Counting Argument

Let G be a group of order $$|G|=pq^m$$ where $p$ and $q$ are primes and $q^m<p$. Show that $$G\cong C_p \rtimes_h Q$$ where $Q$ is a group with $|Q|=q^m$ and $h:Q\rightarrow Aut(C_p)$ is a ...
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1answer
12 views

Abelianization of the absolute group and maximal abelian extension

Let $K$ be any field, $\overline K$ is the separable closure of $K$ and $K^{ab}$ is the maximal abelian extension of $K$. I want to prove the following relation $$G(\overline ...
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1answer
40 views

Galois group isomorphic to $\mathbb Z$

Does exist an example of a Galois extension $L/K$ such that $\text{Gal}(L/K)\cong \mathbb Z$? Thank you.
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1answer
36 views

How can I prove that G is abelian?

$N$ is a finite normal subgroup of order $n$ of $G$, and $|\operatorname{Inn}(G)|=m$ , $(n,m)=1$ and $[G:N]=p$ a a prime number. How can I prove that G is abelian? Can I use that $G$ is abelian iff ...
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1answer
35 views

Two non-isomorphic groups which are epimorphic images of each other

Let $G$ and $H$ be two groups, I am looking for an example such that $G$ is an epimorphic image of $H$ and $H$ is an epimorphic image of $G$ (i.e. they are both quotients of the other group), but they ...
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1answer
22 views

Group of exponent $2$.

When I have a group $G$ of exponent $2$ and I know that all elements in $G-\{e\}$ are conjugated, is it right that $G$ is of order $2$? My try: For $g,h \in G - \{e\}$ the conjugation assumption ...
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2answers
44 views

Group of finite order where every element has infinite order

An often given example of a group of infinite order where every element has infinite order is the group $\dfrac{\mathbb{(Q, +)}}{(\mathbb{Z, +})}$. But I don't see why every element necessarily has ...
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20 views

Subgroup of square generators. [duplicate]

Let $N$ be a subgroup of $G$ with $N$ being generated by $\{x^2|x \in G\}$. Prove that $N$ is a normal subgroup of $G$. And that $[G, G] \subset N$. My idea was to look at $G/N$ and take $f:G ...
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1answer
32 views

Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
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33 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
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3answers
61 views

A line avoiding an Algebraic group

Let $K$ be an algebraically closed field, and $G\subset (K,+)^3$ an algebraic subgroup (i.e. given as the zero sets of finitely many polynomial equations) of dimension 1. Is it clear that there is a ...
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1answer
45 views

Centralizer, normalizer, and center of a dihedral group

Let $A := \{1, r, r^2,..., r^{n-1}\}$. Compute $C_{D_{2n}}(A), N_{D_{2n}}(A),$ and $Z(D_{2n})$. So far I figured that all of the rotations are in the centralizer/normalizer, because all rotations ...
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19 views

Set of all permutations on n generating function [duplicate]

Show that $S_n = \langle (1\ 2), (1\ 2\ \ldots\ n) \rangle$ for all $n \geq 2$. I'm not sure how to approach this one.
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1answer
38 views

Prove that $\mu:G\times G \rightarrow G$ is a homomorphism if and only if $G$ is abelian.

Given $\mu:G\times G$ be the operation on a group $G$; that is, $\mu (a,b)=ab$. Prove that $\mu$ is a homomorphism if and only if $G$ is abelian. I have no problem on proving the necessary ...
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1answer
16 views

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$.

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$. Show that the following statements are equivalent: (a) $\gcd(n,k)=1$, (b) the only ...
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19 views

Simplifying a coset

Let G be a group and let $M,N \leq G$ be normal such that $G = MN$. Prove that $G/(M \cap N) \cong (G/M) \times (G/N)$ I have found a solution to this question here: ...
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1answer
30 views

First isomorphism theorem application

Let G be a group with, $N\subset G$ a normal subgroup, And assume that $H$ is a subgroup of $G$, $H\subset G$. Further $HN=G$ and $H\cap N = \{e\}$ . Prove that $H$ generates the cosets of $N$ in ...
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3answers
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If $G$ and $G'$ are two finite group of same cardinal, then $G\cong G'$.

I have to prove that if $G$ and $G'$ are two finites group of same cardinal, then they are isomorphic. Actually, it looks obvious. Suppose $G=\{g_1,...,g_n\}$ and $G'=\{h_1,...,h_n\}$. Does the ...
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1answer
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How to find identity element of a set (under modular) operation?

Question 1) Can the set of $\{0, 1, 2, 3\}$ under the operation of modulo-$4$ addition and multiplication form a group as well as a field ? If yes then how and if not then why ? Question 2) How to ...
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29 views

Converse Lagrange Theorem?

Let $G$ be a non abelian group of order $12$, and $G \not \cong A_{4}$ Then $G$ contains an element of order $6$. How can I prove it? I know that the converse of Lagrange Theorem is not true for ...
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48 views

Proof involving the group of permutations of $\{1,2,3,4\}$.

Let $\sigma_4$ denote the group of permutations of $\{1,2,3,4\}$ and consider the following elements in $\sigma_4$: ...
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1answer
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What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
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1answer
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Doubt about associative property of a group (Abstract Algebra). [on hold]

I am new to abstract algebra and I have a doubt about the associative property. Suppose a set is given, such as $G=\{0,1,2,3,4\}$ under $\pmod{5}$ addition operation and we have to check whether $G$ ...
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$G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f$ has unique fixed point , then any $p$-Sylow subgroup is normal?

Let $G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f(x)=x \implies x=e$ ; then is it true that for every prime $p$ dividing $|G|$ , there is exactly one $p$-Sylow subgroup ...
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1answer
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Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
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Proof that $\operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F)$ denotes a group-isomoprhism.

Definition: Let $A$ be a Basis of $V$, $V$ a $K$ - Vectorspace. $M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} $, where $\Phi_A$ denotes the following function: $n := \dim V, \{x_1,\ldots,x_n\} = A$ ...
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1answer
25 views

Classifying groups of order $6$ using semidirect products

Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this. By Sylow's theorem, there ...
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1answer
61 views

Sylow counting argument; prove G isomorphic to the direct product.

Let G be a group of order $|G|=pq^m$, where $p$ and $q$ are primes with $q^m<p$. i) Use a Sylow counting argument to show that $G\cong C_p\rtimes_hQ$ where Q is a group with $|Q|=q^m$ and ...
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3answers
33 views

Prove that the four roots of unity form an abelian multiplicative group

My question is: Let $i = \sqrt{-1}$. Prove that the four roots of unity $\{1, -1, i, -i\}$ form an abelian multiplicative group. I know that abelian group is a group with commutative property. ...
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0answers
37 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
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0answers
58 views

On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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3answers
54 views

Order $n$ elements of infinite groups of finite exponent $n>2$

I want to show or to disprove the following result: If $G$ is an infinite group, $n$ the exponent of $G$ is finite, $n>2$, then there are infinitely many elements of order $n$ and infinitely ...