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)

0
votes
0answers
24 views

Result on direct product of groups

Suppose that $G=A \times B$ is finite group as a direct product of $A$ and $B$ and let $U \leq G$. If $N \leq U$ for all $N\unlhd A$ then $A\subseteq U$. I start by letting $a\in A$ be arbitrary. If $...
1
vote
2answers
25 views

Subgroup of order 12

Question: Suppose that $H$ is a normal subgroup of a group $G$. If $\left | H \right |=4$ and $gH$ has order $3$ in $G/H$, find a subgroup of order $12$ in $G$. By the property of cosets: $\left |...
0
votes
1answer
22 views

Number of subgroups of groups with prime power order

Let $G$ be a group of order $p^2$ and put $\mathcal A=\{U\leq G, \#U=p\}$. What is $\#\mathcal A$? If $G$ is cyclic, then $G$ is generated by some element $x$ of order $p^2$. It seems like there ...
-2
votes
0answers
28 views

When is showing any elements in a subset of a group is a sufficient condition for a subgroup. [on hold]

Let G/N be an abelian factor group. To show that H is a subgroup of N, it suffices to show that H is a subset of N. Why is it not necessary in this case to demonstrate that the group axiom holds or ...
2
votes
1answer
36 views

What does extend a group by a map mean?

I have run into a statement in literature and cannot figure out its meaning. The statement is: if $0\to Z^{n}\to G\to K\to 1$ (equivalently $G/Z^{n}=K$), group $G$ is said to be an extension of $Z^{n}...
0
votes
1answer
36 views

A normal subgroup $N$ of $G$ with $\operatorname{gcd}(|N|,|G/N|)=1$ [on hold]

Let $G$ be a finite group and $N$ be a normal subgroup of $G$ such that the centrilizer of $x$ in $G$ is a subset of $N$ for each $x \in N \setminus \{e\}$ ($\operatorname{C}_{G}(x) \subseteq N$, $\...
0
votes
2answers
106 views

Why is identity element required for groups?

I would like to know the necessity for having an identity element for every group. I know the meaning of an identity element.
2
votes
1answer
26 views

Product with a maximal subgroup

Suppose that $G$ is a finite group and $M$ is a maximal subgroup of $G$. If $N$ is a subgroup $G$ not contained in $M$, then $G=MN$. I know $M \subseteq MN$. I want to use the maximality of $M$ but $...
3
votes
1answer
39 views

A group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?

Is it true that a group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?
2
votes
3answers
69 views

If $H$ is a subgroup of $G$ of index $n$, then $g^{n!} \in H \ \forall g \in G$

Let $G$ be a finite group and $H$ a subgroup of $G$ of index $n$, i.e., $[G:H]=n$. Prove that $$\forall g \in G,\; g^{n!} \in H.$$ This is a question I've had in a past exam for Group Theory and I'm ...
1
vote
2answers
55 views

Cyclically reduced words

This is just a reference request. I'm trying to find out whether there are some well developed notes/theory out there (books and the like) focusing on cyclically reduced words in groups. Quickly ...
1
vote
1answer
28 views

$H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then are $H,K$ equal or atleast isomorphic?

Let $H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then is it true that $H=K$ ? Or atleast $H \cong K$ ? ( If $G$ were finite then it would be trivially true ...
1
vote
1answer
39 views

To show that the endomorphism ring of any locally cyclic group is commutative

Let $G$ be a locally cyclic group , then this wiki page https://en.wikipedia.org/wiki/Locally_cyclic_group claims that the endomorphism ring $End(G)$ is commutative , but I am unable to see how it is ...
3
votes
1answer
56 views

How do we call a map $F$ such that $F(g\cdot p)=\varphi(g)\cdot F(p)$?

Let $G$ and $H$ be groups acting on sets $M$ and $N$. Suppose that there is a group homomorphism $\varphi:G\to H$ and a map $F:M\to N$ such that $$F(g\cdot p)=\varphi(g)\cdot F(p)$$ for all $p\in M$ ...
4
votes
1answer
83 views

If $G$ is a locally cyclic group , then is $\operatorname{Aut}(G)$ abelian?

Let $G$ be a locally cyclic group, then is it true that $\operatorname{Aut}(G)$ is abelian ? I know that $G$ has to be abelian but I cannot decide for $\operatorname{Aut}(G)$.
1
vote
2answers
142 views

About an article regarding free groups

I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let $G$ be a non ...
1
vote
1answer
30 views

Class equation and orbit stabilizer theorem

I was reading the proof of the following theorem but I cannot understand how to use the class equation as he wants me to. Theorem Suppose that $G=HK$ where $H$ is a normal locally finite $p'$-...
2
votes
1answer
69 views

$A$ a subset of a finite group $G$ with strictly more than $|G|/2$ elements. Show $AA=G$. [closed]

The question asks (a) Let $A$ be a subset of finite group $G$ with strictly greater than $|G|/2$ elements. Show $AA=G$ and (b) Show this can fail in a monoid. I've been working on this for awhile ...
4
votes
1answer
37 views

If $H<G$ is abelian, and $\chi(1)=[G:H]$ for irreducible $\chi$, then $H$ contains a nontrivial normal subgroup?

Suppose $\chi$ is an irreducible character of a finite group $G$, and $H$ is a nontrivial abelian subgroup such that $\chi(1)=[G:H]$. Why does $H$ contain a nontrivial normal subgroup? I understand ...
1
vote
1answer
42 views

Classifying space of $GL_{n}(\mathbb{F})$?

I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage ...
0
votes
1answer
27 views

Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
1
vote
2answers
79 views

A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
0
votes
0answers
29 views

Representation of $A_5$

Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group $A_5$ is related to the reflection group $H_3$ or the Icosahedral group ? Thanks
-1
votes
1answer
54 views

Need help in understanding the example from Dummit & Foote text

This is an example from Dummit & Foote text. Let $D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>$.Since [$r,s$]$=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
2
votes
1answer
30 views

Index for p-adic subgroups

I always got some problems in computing precisely and understanding indexes for congruence-like subgroups. My problem seems quite simple: what is the index of $(1 + p^r)^2$ in $\mathbf{Z}_p^\times$ (I ...
6
votes
2answers
478 views

What is the correct definition of a group?

What is the correct definition of a group? More precisely the predicate "being a group"? According to Wikipedia A group is a set, G, together with an operation • (called the group law of G) that.....
0
votes
0answers
25 views

Structure of automorphism group

$\def\Aut{{\rm Aut}\,}$ Calculating the automorphism group is very hard to do. Sometimes, your group will satisfy some kind of structure which allows you to determine the automorphism group easily. I ...
-1
votes
0answers
35 views

Discrete Mathematics, Equivalence Relations [closed]

I'm struggling to understand the methodology and logic in proving that a relation is reflexive. If I have $f:A \to B$ as a surjective map. How do I prove that the relation $a \sim b$ is reflexive? I ...
3
votes
2answers
575 views

H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.

Question: Let G be a group and let G' be the subgroup of G generated by the set $S=\left \{ x^{-1}y^{-1}xy \mid x,y \in G \right \}$ $\space$ Prove that G' is normal in G. Solved $\...
-1
votes
0answers
22 views

Converse of Lagrange Theorem for Abelian Groups and Normal subgroup of a particular type?

Suppose $G$ is a group with $o(G) = p_1^{a_1}\cdots p_k^{a_k}$. Any divisor $d$ of $o(G)$ is of the form $p_1^{b_1}\cdots p_k^{b_k}$. Separate powers of $p_i$ and by Sylow's theorem I know their exist ...
0
votes
1answer
21 views

If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
1
vote
1answer
35 views

Related to multiplicative subgroup of positive real line

Let $F$ be a subgroup of the multiplicative group $\mathbb R^*_{>0}$ such that $F$ is dense in $\mathbb R^*_{>0}$, $$N\cap F=\emptyset\ \text{ and }\ NF=N,$$ in which $N$ is a subset of $\...
1
vote
1answer
26 views

Question on cosets and Lagrange's Theorem

If $H$ is a subgroup of $G$ and $|G| = n |H|$ where $n$ is a positive integer, how can I prove that there is some positive integer $k$ with $ 1 \le k \le n$ such that $x^k$ is in $H$ where $x$ is an ...
0
votes
0answers
21 views

Wallpaper groups

I am trying to understand the wallpaper groups and their symmetries. For example consider the following tiling : I believe that the symmetries are two translations, a $120^\circ$-rotation and a ...
1
vote
1answer
26 views

Non-simplicity of groups of order $p^{a}(p+1)$, $p$ a prime

Alperin and Bell, Groups and Representations, section 7, exerc. 10 (a), p. 71, is as follows : if $p$ is a prime number, if $G$ is a group of order $p^{a}(p+1)$, with $a > 1$, then $G$ cannot be ...
0
votes
0answers
17 views

Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
0
votes
0answers
26 views

Graph with small automorphism and large isomorphism

Is there a graph family on $n$-vertices such that any graph $G$ in the family have small automorphism group (say $|Aut(G)|\leq n^c$ for some fixed $c>0$) which if $G$ and $H$ are isomorphic then ...
0
votes
1answer
26 views

isomorphic groups related in direct product.

I want to prove that the direct product $G\times H$ of two groups has a subgroup isomorphic to $G$ and a subgroup isomorphic to $H$. How I thought to prove is that taking a pair $(g,h)$ from $G\times ...
1
vote
1answer
48 views

Is there a general strategy for identifying the automorphism group of a graph?

I understand what an automorphism is, and I can sort of wrap my head around the idea that the set of automorphisms under composition form a group, but when asked to actually find the automorphism ...
0
votes
2answers
29 views

Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$ (12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z} $$ Is this the best answer I can ...
0
votes
0answers
20 views

are there $E(x,G)$ and $E(H,G)$ two subgroups of $G$? [closed]

let $G$ be a group and $H$ be a subgroup of $G$. let $E(x,G) = \{g \in G ; [g,x,x]=1 \}$ and $ E(H,G) = \{ g \in G ; [g,x,x] = 1 ~~~ \forall x \in H \}$ . are there $E(x,G)$ and $E(H,G)$ two ...
0
votes
1answer
27 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
0
votes
1answer
46 views

Group action on cosets

I would like to solve the Problem 2.19 from A Course in Modern Mathematical Physics by Szekeres. The problem is part of the paragraph 2.6 Group action. The formulation is: Problem 2.19 If $H$ is ...
2
votes
1answer
34 views

Can someone explain this proof of the relationship between chromatic number and independence number to me?

I came across the following claim and proof in this paper, and I really don't follow. If $G$ is a vertex-transitive graph with independence number $\alpha$ and chromatic number $\chi$ then $n/α(G) ≤...
35
votes
11answers
2k views

Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
2
votes
0answers
40 views

Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
2
votes
0answers
39 views

To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$ \bigcap_{N \in \mathcal{N}} N \ = \ \{ e\} $$ I know how free ...
1
vote
1answer
34 views

Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $ S^1 \times S^1 \times S^1 $? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
0
votes
2answers
36 views

Showing $NH$ is a subgroup of $G$

Question : If $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$, prove that $NH$ is a subgroup of $G$. Thread is constructed on a mobile so I will attempt to be as succinct as possible. ...
0
votes
2answers
45 views

A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial.

A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial, i.e., $\ f(\sigma) = 0 \ \forall \sigma \in S_{n}$. I started off by thinking that it was something to do with the ...