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.

learn more… | top users | synonyms (2)

1
vote
0answers
29 views

Aut(G) is abelian

I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ...
1
vote
1answer
23 views

$a,b,N$ are integers. Prove $x=x_0+\cdots$, $\ \ y=y_0+\cdots $ are solutions to $ax+by=N$

I'm asked to prove that if $a,b,N$ are integers, then in the equation: $$ax+by=N$$ I must prove that the integers $$x=x_0+\frac{b}{d}t,\ y=y_0-\frac{a}{d}t$$ are solutions to the equation. where ...
11
votes
3answers
125 views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
2
votes
1answer
60 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
0
votes
1answer
14 views

Classifying the central product HK of two cyclic groups

Let group $H$ be a direct product of cyclic groups $C_1$ and $C_2$ of order $p$ and $p^2$ respectively. Let $D=\{x\in H\mid \text{ord}(x)\leq p \}$. D is generated by $C_1$ and subgroup $E$ of $C_2$ ...
0
votes
0answers
13 views

Finding efficiently finite groups whose set of commutators is not a subgroup

The set of commutators of a group might not be a subgroup of the group. I give here such an example from P.J. Cassidy. It is an infinite group. It is possible to derive from this example one of a ...
0
votes
1answer
20 views

all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
1
vote
2answers
56 views

The center of a group $G$ is a subgroup of $G$ [duplicate]

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 ...
0
votes
0answers
33 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 ...
5
votes
3answers
72 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?
0
votes
1answer
24 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 ...
-1
votes
4answers
33 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?
1
vote
1answer
25 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 ...
0
votes
0answers
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 ...
1
vote
1answer
21 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, ...
2
votes
0answers
17 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 $1 \leqslant k \leqslant n$. ...
1
vote
2answers
34 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, $b^{-1}a^{-1}ab = e$. ...
2
votes
3answers
49 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 $kHk^{-1} = H$ ?
10
votes
0answers
69 views

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 ...
-3
votes
0answers
18 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 $ ...
0
votes
2answers
43 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 $(\Bbb Q\setminus\{0\},\cdot)$?
6
votes
2answers
86 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 ...
0
votes
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 ...
-2
votes
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?
4
votes
3answers
51 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. ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
-2
votes
0answers
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 $\ker(nat_{H})$.
1
vote
1answer
43 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 $. ...
1
vote
2answers
96 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 $H/N\cong G$?
0
votes
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 $\left\vert G/N\right\vert =n$?
0
votes
2answers
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 ...
1
vote
1answer
26 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 ...
1
vote
1answer
38 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 ...
1
vote
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 ...
2
votes
1answer
62 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 ...
-5
votes
0answers
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,
6
votes
3answers
454 views

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 ...
8
votes
3answers
179 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 ...
3
votes
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
votes
0answers
83 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 ...
-1
votes
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 ...
1
vote
1answer
47 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
-2
votes
1answer
33 views

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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
-2
votes
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
votes
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 ...