Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [group-theory]

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 is the study of groups.

0
votes
0answers
6 views

On the difference set of all primes is $\Bbb{Z}$, elementary observation.

Let $\Bbb{P}$ be the set of prime numbers in $\Bbb{N}$. Well known open problem: $$ \Delta\Bbb{P} := \{ p - q : p,q \in \Bbb{P}\} = \Bbb{Z} $$ If $2n$ doesn't occur, then adjacent gaps $\{ n\}, \...
1
vote
2answers
40 views

Show that the group $G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$ has order $27$.

This is Exercise 1.2.21 of Magnus et al's book on combinatorial group theory. The Question: Show that the group $$G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$$ has order $27$...
0
votes
0answers
14 views

What surfaces(manifolds) can be the boundary of hyperbolic groups?

Question: What surfaces can be the Gromov boundary of hyperbolic groups? You could also ask the same question except for higher dimensional manifolds. I know that spheres appear as the boundary of ...
0
votes
2answers
21 views

$AB\leq N_G(B)$ implies $B\trianglelefteq AB$?

I am reading the proof of the Second Isomorphism Theorem on Dummit and Foote's Abstract Algebra, 3rd edition. Could anyone tell why they mention $AB\leq N_G(B)$ on the highlighted part? Does $AB\leq ...
0
votes
1answer
20 views

Composition series for dihedral grup of order $2pq^2$

Let $p,q$ be different primes and $D_{2pq^2}=\langle ab\mid a^{pq^2}=b^2=1, ba=a^{-1}b\rangle$ the dihedral group of order $2pq^2$. Find a composition series for $D_{2pq^2}$. I don't know how to ...
6
votes
2answers
634 views

Is there a proof that performing an operation on both sides of an equation preserves equality?

so I was learning some abstract algebra and group theory, when they went over the proof of the cancellation law $$ ab = ac\implies a^{-1}(ab) = a^{-1}(ac)\implies (a^{-1}a)b = (a^{-1}a)c\implies eb=...
2
votes
3answers
80 views

If $ab = e$, then $ba = e$?

$x\cdot (a\cdot b) = x\cdot e$ $(x\cdot a)\cdot b = x$ $e\cdot b = x$ $b = x$ Are my steps correct? What I wanted to prove is that if $ab = e$, then $ba = e$ $x$ is inverse of $a$ and $e$ is ...
2
votes
5answers
65 views

If $a\cdot b = e$ and $x$ is inverse of $a$ , does this imply that $x = b$?

$a,b,x$ are elements of a group . $x$ is the inverse of $a$. Here is my attempt to prove it :- $a\cdot b = e$ $x\cdot (a\cdot b) = x\cdot e$ $(x\cdot a)\cdot b = x$ $e\cdot b = x$ $b = x$ Are ...
0
votes
1answer
44 views

How to represent the groups symbolically?

What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.) 1). Cyclic group of order $p^2 q$. 2). Semidirect product of cyclic ...
0
votes
1answer
38 views

Find the derived subgroup of $A_4$

Find the derived subgroup of $A_4$. Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$. The commutator of ...
2
votes
1answer
81 views

Proving $U=U$ is derivable from any set of relators.

This is Exercise 1.1.1 of Magnus et al's book on combinatorial group theory. The details: Let $G=\langle X\mid R\rangle$ for sets $X, R$. Definition 1: The empty word $\varepsilon$ and the words $...
1
vote
3answers
44 views

$hKh^{-1}=K$ if and only if $h^{-1}Kh=K$?

Suppose $G$ is a group and $K$ is a subgroup of $G$. If $h\in G$, is it true that $$hKh^{-1}=K \Leftrightarrow h^{-1}Kh=K?$$ I am reading a proof on page 94 of Dummit and Foote's Abstract Algebra, ...
5
votes
2answers
51 views

If $N/M$ is a normal subgroup of $G/M$ then $N$ is a normal in $G$

Is it true that if $N/M$ is a normal subgroup of $G/M$ then $N$ is a normal subgroup of $G$ itself? I would think that not necessarily because I would expect that under conjugation we could get a ...
3
votes
1answer
36 views

Injective homomorphism whose image is contained in the union of two conjugacy classes

Show that there exists injective homomorphism $\tau : \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \rightarrow S_{27}$ whose image is contained in the union of two conjugacy classes of $S_{27}...
-4
votes
0answers
60 views

showing G_3 = 1 for Group 147 in Hall & Senior from small number of relations [on hold]

We know that Group 147 in Hall & Senior has Class 2 for the lower central series, that is $G_3 = 1$. Sag & Wamsley give a small number of generators for this group as follows: let $G = \...
1
vote
3answers
50 views

How many elements of order $n$ does $\bigcup_n^\infty\{e^{\frac{2ik\pi}{n}}:0\leq k\leq n-1\}$ have?

I came across a question which I though I managed to solve. After looking at the official solution for it I got very confused. The Question $H=\bigcup_{n=2}^\infty\{e^{\frac{2ik\pi}{n}}:0\leq k\leq ...
6
votes
1answer
81 views

Why does $A_5$ have $\binom{5}{4}$ Sylow 2-subgroups?

Let $Syl_p(G)$ be the number of Sylow $p$-subgroups in a group $G$. Why does $|Syl_2(A_5)|=\binom{5}{4}$? Is this true in general, i.e., does $|Syl_2(A_n)|=\binom{n}{2^\alpha}$ where $2^\alpha$ is the ...
1
vote
1answer
23 views

Product of a subgroup with index 2 with another is the whole group

Let $H,K\le G$ be two subgroups of $G$ with $K\not\le H$ and $[G:H]=2$. Then, is $HK=G$? I think yes, but am stammering in the proof. The index of $2$ suggests normality and hence, that maybe $HK$ ...
1
vote
1answer
90 views

Can one prove that a group of order 150 is not simple using element counting?

I'd like to know whether it's possible to show that a group of order 150 is not simple using only element counting (or mostly element counting). I have seen a solution to this problem here (Every ...
0
votes
0answers
19 views

Does the action of the free abelian group $\Bbb Z^2$ generalise to $\Bbb Z^{n+1}$ on $\Bbb Z\left[\frac1{\prod_n p_n}\right]\setminus0$?

Does the action of the free abelian group $\Bbb Z^2$ generalise to unlimited many dimensions: $\Bbb Z^{n+1}$ on $\Bbb Z\left[\dfrac1{\prod_n p_n}\right]\setminus0$? First consider the following ...
-1
votes
2answers
45 views

Is the direct sum of $p$ copies of $\mathbb{Z}$, $\bigoplus_{p} \mathbf Z $ isomorphic as group to direct sum of $p-\{2\}$ copies

Is the direct sum of $p$ copies of $\mathbb{Z}$, $\bigoplus_{p} \mathbb Z $ isomorphic as group to direct sum of $p-\{2\}$ copies of $\mathbb{Z}$ By $p$, I mean indexing over set of all primes. I ...
5
votes
1answer
48 views

A group with an infinite cyclic normal subgroup that has a finite cyclic quotient is abelian

Let $G$ be a group with a normal subgroup $N$ such that $N$ is isomorphic to $\mathbb{Z}.$ Also suppose that $G/N$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ for some integer $n \geq 2.$ I need to show ...
2
votes
0answers
76 views

How can I learn about the Monster group?

There are several questions about the Monster group on this site, but none really answer the question in the title. While reading about groups in a first year algebra course, I was told about the ...
3
votes
1answer
29 views

Show that $y=x^{k}$ with $gcd(k,n)=1$ is a generator of $G$.

Could someone please verify whether my solution is okay? Let $G$ be a finite cyclic group with $|G|=n$ and generator $x$. If $y=x^{k}$ and $gcd(k,n)=1$, then show that $y$ is a generator of $G$. ...
3
votes
1answer
31 views

On a characterization of quasi-isometric embedding for groups

This question is about the proof of theorem 6.13.12 in Cellular automata and groups, Ceccherini. Let $G,H$ be finitely generated. We say that $\phi:G\to H$ is a quasi-isometric embedding if i) For ...
2
votes
2answers
89 views

Understanding a Cyclic Group Proof

For one of my questions, the following answer was given by Suzet: The thing is: I do not see why... Is it because of inverses or because having a negative could make $i-j$ positive? I asked this in ...
1
vote
0answers
38 views

Quotienting out by a normal subgroup to get a Perfect Group

For the first part, If you have a homomorphic image that is a solvable group, then from the solvable group, we can induce another homomorphism which has a homomorphic image that is abelian. Solvable ...
0
votes
1answer
33 views

Is the group generated by $f(x)= x+2^{\nu_2(2x)}$ and $g(x)=2x$ a 2-dimensional affine space on $\Bbb Z[\frac12]\setminus 0$?

Does the free abelian group generated by composition of $f(x)=x+ 2^{\nu_2\left(2x\right)}$ and $g(x)=2x$ define a 2-dimensional affine space on $\Bbb Z\left[\frac12\right]\setminus0$? $2^{\nu_2(x)}$ ...
-2
votes
1answer
45 views

Solvable, non-nilpotent group with nilpotent commutator subgroup

What is the smallest example of a finite solvable, non-nilpotent group $G$, such that its derived subgroup $G'$ is nilpotent, but not abelian?
7
votes
1answer
35 views

Given a projection $p$, find the largest subset $G(p) \subset Map(S,S)$ that is a group under composition

I want to show, that given a set $S$ and a projection $p: S \rightarrow S$ (i. e. $p = p \circ p$), there exists a largest subset $G(p) \subset Map(S, S)$, which contains $p$ and is closed under ...
-1
votes
1answer
27 views

Group of even order contains an element of order $2$ (Explanation) [duplicate]

I am working on the problem: If G is a group of order 2n, show that the number of elements of G of order 2 is odd. which already has solution (in the link below), please explain the solution in ...
2
votes
0answers
48 views

What interesting properties does $A^3$ have?

Suppose $G$ is a finitely generated group. Suppose $A^1$ is a finite subset of $G$, such that $\langle A^1 \rangle = G$. Let’s define $A^n \subset G$ for $n \in \mathbb{N}$ using the recurrent ...
5
votes
0answers
83 views
+50

Irreducible 2-Brauer characters of $S_5$

Beginning with the ordinary character table of the symmetric group $S_5$, one immediately gets the following Brauer characters in characteristic two: $\begin{array}{c|c|c|c} S_5 & () & (...
2
votes
1answer
57 views

Isomorphism theorem in categories?

In group theory, we know that a group homomorphism $f:G \to H$ induces an isomorphism of groups ${G \over \ker(f)} \simeq \operatorname{im}(f)$. I've searched a lot for a categorical version of this ...
2
votes
1answer
49 views

First Sylow Theorem proof

Let $G $ be group such that $p^a $divides $|G|$ then G has subgroup of order $p^a|$. Proof:Let $|G|=p^am$ Let $\mu$ is set of all subset with $p^a$ elements. SO there are $\binom{p^am}{p^a}$ element ....
1
vote
0answers
47 views

Help verify my understanding of groups?

I just learned the definition of groups today an would like help in verifying my understanding. Please point out any mistakes or misconceptions in my examples. Definition: A group is a pair $G = (G,*)...
1
vote
0answers
41 views

Classify all groups of order $3\times 5\times 67$

I'm working on classifying all groups of order $3*5*67= 1005$ and I've made some progress but I'm not fully there yet. First, I've found that the sylow-67= $S_{67}$ subgroup is normal by using ...
2
votes
1answer
52 views

Show that $GL_n(\mathbb{F}_{23})$ has subgroup of index $2$

Show that $GL_n(\mathbb{F}_{23})$ has subgroup of index $2$ We know that $|GL_n(\mathbb{F}_{23})|=(23^n-1)(23^n-23) \dots (23^n-23^{n-1})$ and therefore is even. I thought about Cauchy theorem ...
6
votes
1answer
35 views

What is the maximal $m$, such that $\mathbb{Z}_2^m \leq GL(n, 2)$?

Is there any closed formula for the function $m(n)$, that is defined as the maximal $m$, such that there is $GL(n, 2)$ has a subgroup isomorphic to $\mathbb{Z}_2^m$? The only things I know currently, ...
-2
votes
3answers
41 views

How do i prove that inverse mod n belongs to the group and can be less than n? [on hold]

ax + ny = 1 proves that x is inverse of a . But how do i prove that x belongs to the group and can be less than n ?
1
vote
2answers
59 views

How many distinct composition series does the group $D_{12}$ have?

How many distinct composition series does the group $D_{12}$ have? I know that $D_{12} \trianglerighteq \mathbb{Z}_6 \trianglerighteq \mathbb{Z}_3 \trianglerighteq \{e\}$ is a composition series ( ...
3
votes
1answer
34 views

Cancellation in a presentation of a group

In a method for presenting a subgroup we are given A group $$\ G= \bigl\langle\, x, \ y \mid x^2 yxy^3 , \ y^2 xyx^3\,\bigr\rangle$$ So $$\ G/G' = \bigl\langle\, x, \ y \mid x^2 yxy^3 , \ y^2 xyx^...
2
votes
0answers
37 views

How many solvable configurations of the Rubik's cube have no two squares of the same color touching?

The Superflip (image below) is an example of a configuration where no two squares of the same color are adjacent. I'm interested to know how many solvable configurations (that is, those that you can ...
3
votes
1answer
40 views

Which is the value of $t$ ?

It is given $t\in \{0,1,2,3,4\}$, a group $G$ and an element $a$ of $G$ such that $a^8=a^t$ and $a^{88}=a^{74}$ in $G$. If $a^2\neq a^0$ which is $t$ ? For $a\neq 0$ we have the following: \begin{...
0
votes
1answer
25 views

What do you call subsets of $G$-sets whose elements are permuted under some action $G$?

Given any group $G$ acting on any set $X$ via some left or right action $\varphi:G\times X\to X$ is there a name for subsets $Y\subseteq X$ with the property that for any $g\in G$ wge have $\{\varphi(...
0
votes
1answer
38 views

Every group of order $2p^n$, when p is prime, is solvable

I have to prove that every group of order $2p^n$, p is prime, is solvable. When $n=1$, the group either is cyclic or dihedral, in any case the group is solvable. When $n=2$, if P is p-sylow subgroup,...
1
vote
1answer
41 views

Problem from Herstein's Topics in Algebra

Let $G$ be a group in which, for some integer $n > 1$, $(ab)^n = a^nb^n$ for all $a, b ∈ G$. Show that (a) $G(n) = \{x^n | x ∈ G\}$ is a normal subgroup of $G$. (b) $G(n- 1) = \{x^{(n- 1)} | x ∈ G\}...
2
votes
1answer
34 views

Understanding Proof of Sylow theorem from Herstein

I was trying to learn group theory indepedently form Herstein' Topics of Algebra. I that I now reading Sylow's theorem.In that I trying to understand first proof of sylow theorem. But I am not able to ...
0
votes
0answers
35 views

Finite group with cyclic Sylow-2 group has a normal subgroup of index 2. [duplicate]

I'm working on the following qual problem and I'm not sure how to do parts b and c. a) This uses a nifty argument with considering the map $N_G(S) \to \operatorname{Aut}(S)$ via conjugation. And ...
0
votes
1answer
48 views

A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple. [duplicate]

Let $G$ be a finite group of even order, whose $2$-Sylow subgroups are cyclic. Show that $G$ is not simple. I was trying to use Cayley theoem, place $G$ in some $S_m$ and get a contradiction with $...