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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The center of a group G is a subgroup of G

Definition for the center of a Group: The center Z(G) , of a group G is the subset of elements in G that commute with every other element of G. Theorem: The center of a group G is a subgroup of G. ...
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25 views

I want to know if the below sentence is true and why?

I want to know if the below sentence is true and why? Let $G$ be an insoluble finite group then there exists $\pi\subset\pi(G)$ such that if $K=O_{\pi}(G)$ and $\bar{G}=G/K$ it follows that ...
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3answers
55 views

An example of a group such that $G \cong G \times G$

I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?
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1answer
21 views

Examine if $\phi$ is a homomorphism and determin $\ker \phi$ and $Im \phi$.

Let $G=(\mathbb C^*,\cdot), G'=(\mathbb R^*,\cdot)$ and $\phi : G\to G'$ be defined by $\phi(z)=|z|, z\in \mathbf C^*$, where $\mathbf C^*=\mathbf C-\{0\}$ and $\mathbf R^+$ is a set of all positive ...
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4answers
30 views

If $A$ and $B$ are subgroups of a group $G$, is their product also a subgroup of $G$? [on hold]

Is there a theorem to show the above holds truism? how does one show?
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13 views

Characters of transitive finite permutation group

I know that Frobenius reciprocity helps us to solve this problem, but I don't know why: Let $ G $ be a transitive finite permutation group with permutation character $ \pi $. If $\chi $ is an ...
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12 views

Two-step subgroup test “IFF” condition

The theorem for the two-step subgroup test says: The subset H of a group G is a subgroup IFF the binary operation of 2 ordered pairs of elements of H are in G and for each element in H, there each ...
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1answer
18 views

One-to-ones-ness of Group Elements

I came along this theorem in a book, saying that if $a,b,c \in G$, where $G$ is a group, then if $ab = ac$, then $b=c$. It looks like if we assume that all elements in the set are functions (because, ...
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16 views

What is $Hom((S^1)^k , (S^1)^n)$?

I am trying to find $Hom_{gp}((S^1)^k , (S^1)^n)$ , which is the set of continuous group homomorphisms from the $k$ dimensional torus to the $n$ dimensional torus where $1 \leqslant k \leqslant n$. ...
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2answers
32 views

Groups - Proof that $(ab)^{-1} = b^{-1}a^{-1}$

I read this proof in a book: Prove that if $a, b \in G$, then $(ab)^{-1} = b^{-1}a^{-1}$. Proof: Let $a,b \in G$. Then $abb^{-1}a^{-1} = aea^{-1} = aa^{-1} = e$. Also, $b^{-1}a^{-1}ab = e$. ...
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47 views

$G$ a group and $H$,$K$ subgroups, $kHk^{-1} \subseteq H \implies kHk^{-1} = H$?

As post said, if $G$ a group and $H,K \leq G$ and for FIXED $k \in K$ does $kHk^{-1} \subseteq H$ imply that $kHk^{-1} = H$ ?
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Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
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17 views

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. [on hold]

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. is there theorem that said can let $ K = K_{\pi^{\prime}}K_{\pi^{\prime}}$ that $K_{\pi}$ is $\pi$-Hall subgroup of $ K $ and $ ...
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2answers
40 views

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$? [duplicate]

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?
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2answers
85 views

Elementary question in Group Theory with less prerequisite

Here I am posing a problem, which my beginning students of algebra were discussing for long time. Question: Without using theorem of Cauchy or Sylow, can we show that a group of order $15$ contains ...
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0answers
31 views

Is group theory a generalization of number theory [on hold]

The applications of group theory are abound. Many mathematical objects are examined by associating groups to them and studying the properties of the corresponding groups. But number theory and Graph ...
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1answer
28 views

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $|H|=n|G|$ and $G$ is a normal subgroup of $H$?

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $\left\vert H\right\vert=n\left\vert G\right\vert$ and G is a normal subgroup of H?
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3answers
49 views

How to prove that $(G,*)$ is a group?

Let $G=\mathbb{R_0}\times\mathbb{R}$ where $\mathbb{R_0}=\mathbb{R}\setminus\{{0}\}$. Define operation $*$ on $G$ by $(a,b)*(x,y)=(ax,a^2y+b)$. I'd like to prove that $(G,*)$ is a group. ...
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1answer
35 views

uniqueness of identity of a group G

The theorem for uniqueness of identity of a group says there is one identity element $e$ in a group and this element $e$ is unique. My book states the proof as follows: $a.e=a$ for all $a \in G$ and ...
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2answers
30 views

How to prove that $a^{|G|}=e$ if a $\in G $

How to prove that; $a^{|G|}=e$ if a $\in G $ if $G$ is a finite group and $e$ is its identity. I think this could be done through pigeonhole principle but I don't want to use the Lagrange ...
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1answer
25 views

General question on notations when dealing multiplicative and additive modulo

One of the property for the requirement for a set to be a group is associativity. Under ordinary multiplication: $\large{a(bc)=(ab)c}$ Under ordinary addition: $\large{a+(b+c)=(a+b)+c}$ What ...
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19 views

Is $\ker(nat_{H})=H$ a true statement? [on hold]

Is $\ker(nat_{H})=H$? $nat_{H}$ defines as $nat_{H}(a)=a*H$ I know what $ker$ and $nat_{H}$ are but I am not familiar with $\ker(nat_{H})$.
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1answer
41 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
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2answers
94 views

Is every group isomorphic to some nontrivial quotient group?

For any group $G$, does there exist a group $H$ and a nontrivial normal subgroup $N$ of $H$ such that $H/N\cong G$?
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1answer
24 views

Do there exist nontrivial quotient groups of arbitrary finite order?

For any $n\in \mathbb{N}$, does there exist a group $G$ and a nontrivial normal subgroup $N$ of $G$ such that $\left\vert G/N\right\vert =n$?
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48 views

Is the set $SL(2, \mathbb F)$ an Abelian group?

For the set $SL(2,\mathbb F)$, where $\mathbb F$ are entries from either $$\mathbb{Q},\mathbb{R},\mathbb{C} \text{ or } \mathbb{Z}_p \text{ (p is prime)}$$ How should I start by checking this matrix ...
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1answer
25 views

Complex numbers modulo integers

Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$? Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates ...
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1answer
37 views

When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
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1answer
34 views

How is a symmetric group the subgroup of the group of isometries of three-dimensional space?

So I have this question to solve. I've already shown that the group of rotations of a cube is isomorphic to $S_4$. I need to prove that these two groups are not conjugate when considered as subgroups ...
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1answer
61 views

Group presentation of Integers $\big(\mathbb{Z,+}\big)$

I can't understand how is it possible to represent the group $(\mathbb{Z},+)$ as follows $$\mathbb{Z} = \big<a\big>$$ with only one generator and no relations ? How can there be no relations ...
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61 views

What are $S_{n}$ and $A_{n}$ in group theory? [on hold]

What are $S_{n}$ and $A_{n}$ in group theory, and is $[S_{4},A_{4}]=4$? I know that $S$ has to do with permutations, but I am not sure if thats right. Thanks,
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Why is Rationals w.r.t addition not an Isomorphism to Rationals w.r.t. multiplication?

Question states: Recall the additive groups Z,Q and R, and the multiplicative groups Q* and R* of non-zero numbers. show that: (b) Q is not isomorphic to Q* (c) R is not isomorphic to R* I can see ...
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3answers
150 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices (dots), in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon and no vertice ...
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2answers
74 views

Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.

Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an ...
2
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0answers
81 views

verify that the set $\{0,1,2,3\}$ is not a group under multiplication modulo $4$

Given the set $\{0,1,2,3\}$: -Associativity holds for this set -Closure holds for this set (constructing the Cayley table, all entries in the tables are in this set). -there is an identity element ...
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1answer
38 views

Determine if $(((13)),\circ)$ is a normal subgroup of $(S_{3},\circ)$ [on hold]

Let $((13))$ denote the group generated by $(13)$. Is $(((13)),\circ)$ a normal subgroup of $(S_{3},\circ)$? Also is $(((123)),\circ)$ a normal subgroup of $(S_{3},\circ)$? I have just started ...
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1answer
46 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
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1answer
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Properties of homomorphisms

I have some problems in how to prove these: Let $f$ be homomorphism from group $G$ to a group $N$. Prove the following: $k\le G$ iff $f[k]\le N$ $f$ is onto iff range of $f =N$ $f$ is one-to-one ...
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0answers
19 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
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1answer
23 views

Commutator and upper-lower centers question

Let $H$ be a normal group of a group $G$. $H$ is a subgroup of the $k$-th lower center $\gamma_k(G)$. I have a relation like the following $$ [H,G,G,\dots, G] = 1 \qquad (n\; \text{times} \; G) $$ but ...
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1answer
34 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
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1answer
54 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
3
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1answer
24 views

Generating Constrained Random Distributions

I am trying to help another StackExchange user. We are attempting to fill a 6x6 matrix with 12 A's, 12 B's, and 12 C's subject to the constraint that each row contains 2 A, 2B and 2 C and each column ...
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0answers
30 views

Prove that if $G/Z(G)$ is isomorphic to $\mathbb{Z}_3 ×\mathbb{Z}_3$, then $G$ is isoclinic to an extraspecial group of order $27$. [on hold]

Let $G$ be a non-abelian group. Prove that if $G/Z(G)$ is isomorphic to $\mathbb{Z}_3 ×\mathbb{Z}_3$, then $G$ is isoclinic to an extraspecial group of order $27$.
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26 views

Principal congruence subgroup index in $SL(2,\mathbb{Z})$

Why has the principal congruence subgroup, \begin{equation} \Gamma(N)~=~\Bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL(2,\mathbb{Z})~|~a\equiv d\equiv 1 ~\text{és}~ b\equiv c\equiv ...
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42 views

transitivity of induction

I want to prove the "transitivity of induction" property: Let $ H\leq K\leq G $ where $ G $ is finite. Let M be an $ FH $-module, where $ F $ is any field. Then $ (M^K)^G\simeq^{FG} M^G $. Would you ...
2
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1answer
31 views

Number of elements in Hom$(S_n,\mathbb{C})$

Hox can I determine the number of elements in Hom$(S_n,\mathbb{C})$ for $ n\geq 1$? I thought maybe I can use the thesis that for a normal subgroup $N\subset G$, and a subgroup $H\subset G$, there ...
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1answer
43 views

Let $G$ be a non-abelian group. Prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 × \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$ [on hold]

prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 \times \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$.
2
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0answers
51 views

Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
2
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0answers
39 views

Product of Conjugacy Classes in a Group

Let $G$ be a non-abelian group, and consider $x,y$ in $G-Z(G)$. Let $C(x)=x^G$ and $C(y)=y^G$ denote the conjugacy classes of $x$ and $y$ respectively. Question: What conditions on $x,y$ imply that ...