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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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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10 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
30 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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$G$ 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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16 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
38 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
78 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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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
27 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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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
29 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
24 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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37 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
22 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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47 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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448 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 ...
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3answers
133 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 $k$ dots, in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon and no vertice remains ...
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2answers
73 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 ...
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80 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
45 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
32 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 ...
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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
33 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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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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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 ...
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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 $D_8$.
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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 ...
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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 ...
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1answer
29 views

If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
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1answer
91 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
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1answer
16 views

Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...
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1answer
54 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
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1answer
16 views

Properties of generalized characters

I search for generalized characters which are not characters.Also I want to know that why every generalized character is a difference of characters.
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1answer
38 views

Generalization of Burnside theorem [on hold]

I probe for understanding the proof of Burnside $ p-q $ theorem: a finite group with a nilpotent subgroup of prime-power index is soluble.
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1answer
50 views

How many non-isomorphic groups of order 122 are there?

How many non-isomorphic groups of order 122 are there? Let $G$ be a group of order 122.No of Sylow 61 subgroups of order 61=1 and hence it is normal say it is $H$. No. of Sylow 2 subgroups of order ...