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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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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Classify all finite groups with property

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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1answer
20 views

Order of a $\alpha \beta^{n/q}$ given the order of $\alpha$ and $\beta$

I'm asked to prove the following easy result: Let $G$ be a finite abelian group. Let $\alpha \in G$ of order $m$ and $\beta \in G$ of order $n$. Assume the $n\not\mid m$, and let $q=p^v$ for some ...
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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
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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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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
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When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
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1answer
50 views

how to find coefficient c1, c2, c3, c4 of a polynomials of degree 4 from resolvent

if not starting from standard resolvent of each degree and use (y-x1...)(y-x2...)(y-x3...) and group theory how to find corresponding c1, c2, c3, c4 of polynomial x^4+c4*x^3+c3*x^2+c2*x+c1 which c1, ...
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1answer
22 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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22 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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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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1answer
84 views

Trying to show $|ab|$ divides lcm$(|a|,|b|)$

I'm trying to solve this Putnam problem. The problem is "show that for a finite group with $n$ elements of order $p$, where $p$ is prime, either $n=0$ or $p\: \vert\: n+1$." I'm trying to do this by ...
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0answers
29 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
21 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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Normal semigroups- semilattice of groups!

Can someone please help me to understand this : If $S$ is a Clifford semigroup with the set of idempotents $E$, then $S'$ be a sub-semigroup of $S$ ( so $S$ be a semilattice with the same set of ...
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1answer
8 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
37 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
34 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
29 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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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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1answer
32 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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2answers
43 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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30 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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1answer
185 views

If a group mod its commutator is cyclic, then the group is abelian

If $G/G'$ is cyclic then G is abelian. Let G be a group and suppose $G/G'$ is cyclic.Then for all g in G we have $ g ∈ gG' = x^k G'$ for some integer k. In particular, $g=x^kz$ for some integer k and ...
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0answers
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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Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$

Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$ I have tried to using the definition but failed. Could someone help me ...
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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
475 views

General linear group/special linear group is isomorphic to R*

Let $GL(n,\mathbb{R})$ be the group of invertible $n \times n$ real matrices, let $SL(n,\mathbb{R})$ be the group of $n \times n$ real matrices of determinant $1$, and $\mathbb{R}^*$ be the group of ...
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1answer
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If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is ...
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1answer
43 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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1answer
159 views
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If $N\lhd H×K$ then $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially

I am thinking on this problem: If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially. I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ ...
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Prove that if $G$ is an Abelian group, then for all $a,b$ in $G$, $(ab)^{n} = a^{n}b^{n}$

This question have already been asked on this site, but i could not understand the details so i ask it again. Also what i have done is that first for $n=1$ its trivial, for $n=2$ we have ...
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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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2answers
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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
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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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18 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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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
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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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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3answers
41 views

Prove that G has a normal Sylow p-subgroup

Suppose that $|G| = pq$ where $p$ and $q$ are distinct primes such that $p$ does not divide $q-1$. Prove that G has a normal Sylow $p$-subgroup . I know what by Sylow's Theorem, either $n_p=1$ or ...
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4answers
568 views

To prove that every element of $G$ has finite order.

Let $G$ be a group such that the intersection of all its subgroups which are different from $e$ is a subgroup different from $e$. Prove that every element of $G$ has finite order. Can i get some ...
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162 views

Suppose a finite set $G$ is closed under associative product and that both cancellation laws hold in $G$. Prove $G$ must be a group.

Suppose a finite set $G$ is closed under associative product and that both cancellation laws hold in $G$. Prove $G$ must be a group. I somehow need to prove identity, inverse, that closure holds to ...
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28 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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1answer
588 views

If $G$ has no non trivial subgroups, then show that $G$ must be of prime order.

If $G$ has no non trivial subgroups, then Show that $G$ must be of prime order. This question is from Herstein Page 46 Question 3. Attempt: Let $G$ has prime order(say $p$). By Lagrange theorem, ...
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1answer
36 views

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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1answer
22 views

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
30 views

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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23 views

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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31 views

$G$ 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 ...