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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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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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
36 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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18 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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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
13 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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14 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
28 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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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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15 views

Regular semigroups- normal semigroups!

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 idempotents( $E$) such that for every $e \in E$, ...
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0answers
25 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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35 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
21 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
33 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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22 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 ...
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1answer
21 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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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
21 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
23 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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29 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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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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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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36 views

Hausdorff quotient space

consider a smooth manifold $M$ and a group action, i.e. a group homomorphism $\phi: G\rightarrow S(M)$, where $S(M)$ denotes the group of diffeomorphisms of $M$. Suppose that for all $K\subset M$ ...
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3answers
48 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 ...
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1answer
32 views

Prove that every sylow $2$-subgroup of $G$ is abelian.

Let $G$ be a finite group and $H \unlhd G$ where $|H|$ is odd and $G/H$ is abelian. Let $P$ be a sylow $2$-subgroup of $G$, then can we say that $P$ is abelian?
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Group with > 1 elements that is a free group and a permutation group?

Consider the set $S = \{A, B\}$ and the group $G = \{e_G, \varphi\}$ with operation composition (where $e$ is the identity map and $\varphi$ maps $A$ to $B$ and $B$ to $A$). This is the permutation ...
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1answer
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Calculating $| \langle \cup_{i=1}^n P_i \rangle |$ where $P_i$ are Sylow subgroups of G

I'm trying to prove: Let $\lbrace P_i: i \in I \rbrace$ be a set of Sylow subgroups of a finite group G, one for each prime divisor of $|G|$. Then $\langle \cup_{i \in I} P_i \rangle = G$. (From ...
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2answers
25 views

CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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1answer
61 views

High-school group-theory problem(given in a contest)

Let $G$ be a finite group and let $ H \le G $ be a subgroup of $G$. Suppose there is some $ \emptyset \neq S \subset G$ such that for any $x\in S$ we have $x^2 \notin H$. Prove that there is $T ...
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28 views

Statements about the order of a group

Let $G$ ba group. The statements are equivalents (i) $|G|$ is prime (ii) $G$ does not have a non-trivial proper subgroup. (iii) $G\simeq\mathbb{Z}/p\mathbb{Z} $ A suggestion to prove these ...
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In group theory, do $ \langle\mathbb{Z}, +\rangle $ and $ \langle\mathbb{R}, +\rangle $ have the same order?

Of course, $ \mathbb{Z} $ is countable, while $ \mathbb{R}$ is uncountable, so the two cannot be isomorphic. However, the reason for my question is the notation for the order of $ $$ ...
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Show that Z*n is the maximal subset of Zn which is a group with the operation [a] · [b] = [ab].

Let Zn be the set of equivalence class of integers mod(n) which are relatively prime to n, i.e. Zn = 􏰀[i] | gcd(n, i) = 1􏰁. So I understand how this works when the binary operation is ...
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1answer
37 views

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$ I have no idea how to show this. I'm studying for a test, so I am less interested in solutions and hints than I am strategy. What ...
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Questions on the proof of Beilinson-Bernstein localization theorem

I am trying to understand the Beilinson-Bernstein localization theorem (following the book by Hotta, Takeuchi and Tanisaki). I got stuck at the following two steps. Any help will be greatly ...
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1answer
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$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
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1answer
41 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
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Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
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1answer
45 views

Quotient of direct sum of abelian groups [on hold]

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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How to find an element and a subgroup of a certain given order in $U(n)$

How can I find a subgroup of order $k$ in $U(n)$ or an element of order $k$? Here $U(n)$ is the group of units modulo $n$. For example, if $n=700$ and $k=6$ I know that since $700=5^2 \cdot 7 ...
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1answer
43 views

Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$? [duplicate]

Let $G$ be a finite group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?
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1answer
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Smallest $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$

I solved the following exercise: Find an integer $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$. Here $U(n)$ is the group of units modulo $n$. To solve it ...
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1answer
78 views

If $f:\mathbb{Z} \to \mathbb{Z}$ is an isomorphism, prove that $f$ is the identity map. [on hold]

I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the ...
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1answer
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In $U(55)$ show $x \mapsto x^3$ is injective

I am stuck on showing that $\varphi : U(55) \to U(55) $ given by $x \mapsto x^3$ is an isomorphism. I already knwo that $\psi: U(n)\to U(n), \psi(x) = x^k$ is an isomorphism if and only if $\gcd(k,n) ...
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1answer
22 views

Direct Product and Quotient of Groups

Quick (and basic) group theory question: Say G, H, K some (Lie) groups, does it in general hold that $$ (G \times H)/H = G $$ and that $$ H = K\times G \to K = H / G $$ And if so, does it then ...
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If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$

The following is an exercise from D. Robinson: A Course in the Theory of Groups. Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k ...
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1answer
31 views

Can the galois group be the symmetric group, if the discriminant is a perfect square?

Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square. Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of ...
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24 views

Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
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1answer
65 views

Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
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2answers
32 views

$G$ a finite group such that $x^2 = e$ for each $x$ implies $G \cong \mathbb{Z}_2 \times … \times \mathbb{Z}_2$ ($n$ factors)

Let $G$ be a finite group such that $x^2 = e$ for each $x \in G$. I know already that $G$ is abelian and that the order of $G$ is $|G| = 2^n$ for some $n \geq 0$. Now I wish to show that $$G \cong ...