Stack Exchange Network

Stack Exchange network consists of 175 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

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.

3
votes
3answers
37 views

When is a quotient group $G/H$ abelian?

So clearly if $G$ is abelian and $H$ is a normal subgroup of $G$ then $G/H$ is abelian since $$xH.yH =(xy)H=(yx)H=yH.xH$$ But is there cases when this quotient group is abelian without the group G ...
1
vote
1answer
35 views

Prove that the group of moves of the Rubik’s cube is not abelian.

I'm currently working in the following excercise: Remember that $G$ is the group of moves of the Rubik’s cube. Prove that this group is not abelian. I'm starting from picking two moves $M_1$ and $...
1
vote
0answers
33 views

If $G$ is a non-abelian finite group, then $|Z(G)| \leq \frac {1}{4} |G|$

I know this is question has been asked several times on here, only hints given ,but just want to check if I have the right idea. My attempt: Suppose G is non abelian finite group and $|Z(G)| \gt \...
1
vote
0answers
25 views

Center of the dihedral group with odd and even number of vertices

I have posted a proof below, and would appreciate it if someone could review it for accuracy. Thanks! Problem: Let n $\in$ $\mathbb{Z}$ with $n$ $\ge$ 3. Prove the following: (a) Z(D$_{2n}$) = 1 ...
1
vote
0answers
8 views

Prove that if $H,K \leq G$ where $G$ hyperbolic, $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate

Let $G = \langle S \mid R \rangle$ be a finite presentation, $G$ is $\delta$-hyperbolic. Prove that if $H,K \leq G$ where $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate. I am ...
3
votes
1answer
35 views

$\mathbb{Z}^2 \ast \mathbb{Z}^2$ is isomorphic to no finite index proper subgroup of itself

I want to prove that $G = \mathbb{Z}^2 \ast \mathbb{Z}^2$ is isomorphic to no finite index proper subgroup of itself Here is my partial attempt: Consider the Bass-Serre tree $T$ that $G$ acts on in ...
0
votes
2answers
33 views

Why is the symmetry of an equilateral triangle D3 and not D3h?

On this wiki page, it is explained that the symmetry of an equilateral triangle is $D_3$; the equilateral triangle is symmetric under the operations of $D_3$. The operations for $D_{3h}$ symmetry are ...
0
votes
1answer
29 views

Prove that all proper subgroups of a group of order $8$ are commutative.

Prove that all proper subgroups of a group of order $8$ are commutative. Let $H$ be a subgroup of a group $G$. Then $o(H)\mid o(G)\implies o(H)=1,2,4$. If $o(H)=1 $ or $2$, then $H$ is commutative. ...
0
votes
0answers
18 views

Finding a surjective homomorphism $G \to \mathbb{Z}[\frac12, \frac13]$ which is not iso

Let $G = \langle a,t \mid t a^2 t^{-1} = a^3 \rangle$. I am asked to find a surjective group homomorphism $G \to \mathbb{Z}[\frac12, \frac13] \rtimes \mathbb{Z}$, where $\mathbb{Z}[\frac12, \frac13] :=...
1
vote
2answers
50 views

Show $N$ is a normal subgroup of $G$ where $G$ is a subgroup of $GL_{2}(\mathbb{Q}).$

We have $G= \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \text{with $a$ and $c$ in $\{\pm 1\}$ and $b$ in $\mathbb{Z}$} \right\} $, which is given to be a subgroup of the group of ...
4
votes
2answers
50 views

Definition of infinite cyclic group

I'm having some conceptual issues with the infinite cyclic group $C_\infty$. Finite groups $C_n$ have a clear representation as integers $0,1,\cdots,n-1$ under addition $(\operatorname{mod} n)$, or as ...
5
votes
2answers
133 views

What is this group $G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle$

Consider the group presentation $$G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle.$$ Is this a known group? What is $G$ isomorphic to? Thanks a lot.
0
votes
2answers
26 views

Prove that $H=\{i,(12),(34),(12)(34)\}$ forms a non-cyclic subgroup of $S_4$.

We have to prove that $H=\{i,(12),(34),(12)(34)\}$ forms a non-cyclic subgroup of $S_4$. It is seen that there is no element of order $4$ in $H$. So, $H$ is non-cyclic. But How can I show $H$ is ...
0
votes
2answers
39 views

On Properly Discontinuous Groups

I need to prove the following: Every group of diffeomorphims that act properly discontinuous in a compact smooth manifold is finite. I've been looking for it in some many references but couldn't ...
4
votes
3answers
120 views

Books that leaves proofs for the reader [on hold]

What are some good introductory books that leave many proofs as exercises? I have been self studying analysis by reading Tao's two fantastic books which eventually leaves most of the (easier) proofs ...
-1
votes
1answer
221 views

Is it true that $G/N$ has a presentation $\langle x,y\mid xyxy^{-1} \rangle $? [on hold]

Let $G=F( x,y)$ be a free group with two generators, assume that $H\leq G$ where $H=\langle xyxy^{-1}\rangle $, let $N$ be the normal closure of $H$. Is it true that $G/N$ has a presentation $\langle ...
0
votes
0answers
36 views

Quotient group of matrices and permutation matrices

The overall objective here is to find a subspace that is invariant to invariant to permutations. Specifically, if we are given a matrices $X$ and $Y$, can we define a subspace where $X \sim Y$ if $X =...
5
votes
1answer
45 views

A polynomial algorithm to determine whether a finite group is nilpotent

Does there exist a polynomial (in respect to the order of the group) algorithm that given a Cayley table of a finite group determines, whether a group is nilpotent or not? There do exist polynomial ...
12
votes
3answers
2k views

Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
0
votes
0answers
67 views

Injective order preserving homomorphism $\phi$ of two ordered abelian groups satisfies $a<b\iff \phi(a)<\phi(b)$.

Exercise: Prove that every injective order preserving homomorphism $\phi$ of two ordered abelian groups satisfies $a < b \iff \phi(a) < \phi(b)$. Definition: A homomorphism $\phi \colon G ...
3
votes
1answer
124 views

A theorem in the paper “Noncommuting Random Products” by Furstenberg

I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963. The statement is as follows: Let $\mu$ ...
-4
votes
1answer
25 views

Suppose that $G$ is non-abelian group and $|G|=pq$ such that $p$ and $q$ are prime numbers [on hold]

Suppose that $G$ is a non-abelian group and $|G|=pq$ such that $p$ and $q$ are prime numbers and $N$ is normal subgroup of $G$ such that $|N|=q$ . show that $G'=N$.
4
votes
1answer
80 views

GAP code to calculate the a certain subgroup $E(G)$ of a group

I am a research scholar from India. At present, I am working on a problem. For this problem, I need to construct the subgroup $E(G)$ of a group $G$ in GAP. Please help me. My question is as follows: ...
0
votes
3answers
94 views

Is this a well-known group? $\langle a,b \mid a^5=b^4=e,b^{-1}ab=a^{-1}\rangle$

Consider the group $$ G=\langle a,b \mid a^5=b^4=e,b^{-1}ab=a^{-1}\rangle $$ It looks like a dihedral group but it is not isomorphic to a dihedral group. Is this a well-known group?
0
votes
0answers
33 views

Choice of symbols: $O_p(G)$, $O^p(G)$, and $O_\infty(G)$

For a finite group $G$ and a prime number $p$, several normal subgroups are defined as follows: $O_p(G)$ = the largest normal $p$-subgroup of $G$ ($p$-core) $O^p(G)$ = the smallest normal subgroup $N$...
5
votes
2answers
144 views

Efficient computation of conjugacy classes of a small group.

I'm trying to construct a character table for a group of order 54 given by: $$ \langle a,b : a^9 = b^6 = 1, b^{-1} a b = a^2\rangle $$ To do this first I need to compute conjugacy classes. This ...
1
vote
1answer
131 views

What is the sequence of accumulation points in the 2-adic space, of the Collatz graph?

In the orbit of the function $3x+2^{\nu_2(x)}$ through "accumulation points" of the Collatz graph I have: $?\mapsto\dfrac{-\langle2\rangle\cdot\{5,7\}}{9}\mapsto\dfrac{-\langle2\rangle}{3}\mapsto \...
0
votes
1answer
22 views

isomorphism in ordered monoids

I read that a morphism $\gamma : S \rightarrow T$ is an isomorphism if there exists a morphism $\Psi : T \rightarrow S$ such that $\gamma \circ \Psi = I(T)$ and $\Psi \circ \gamma = I(S)$, where $I$ ...
-1
votes
1answer
28 views

Prove that there is no nonabelian simple group of order less than 60 [duplicate]

Any tips for this question? I don't want the answer itself, just figure out how must I proceed.
0
votes
2answers
38 views

Group action on convex cone

I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: An open subset of $\Omega \subset \mathbb R^k$ is an open convex cone if it ...
1
vote
0answers
36 views

Show that the center of quaternions group $\textit{Q}$ is generated by the unique element with order 2.

$\textit{Q}$ is a group with order $8$, generated by $a,b$ where $a^4=1$, $b^2=a^2$ and $bab^{-1}=a^{-1}$. I already proved that the unique element of $\textit{Q}$ with order $2$ is $a^2$. How can I ...
1
vote
0answers
24 views

$\varprojlim(SL_n(Z)/K_n(p^i))_{i \in \mathbb{N}} \simeq SL_n(Z_p)$ and $\varprojlim(SL_n(Z)/K_n(m))_{m \in \mathbb{N}} \simeq SL_n(\hat{Z})$

Problem. Show that the natural map $\mathrm{SL}_{n}(\mathbb{Z}) \to \mathrm{SL}_{n}(\mathbb{Z}/m\mathbb{Z})$ is surjective, for all $m$ and $n$. Denoting its kernel by $K_{n}(m)$, show that $$\...
0
votes
3answers
39 views

Group Expression Relating to Cosets

I have a simple question about cosets that is evading me if anybody can provide a hand. I don't think the title informs the question much so if anybody can rephrase the title for me that would be ...
1
vote
0answers
71 views

Quadrants of a cyclically ordered group

I am trying to prove that if there are two different positive elements in a non-linearly cyclically ordered group, then every quadrant of the group is not empty. Is this a correct statement? ...
0
votes
0answers
27 views

minimal normal group of a finite solvable group is elementary abelian p-group

Let $G$ be a finite solvable group. Suppose that $H$ is a minimal normal subgroup of $G$. Then we can raise $H$ to a composition series since $G$ is finite. Since $G$ is solvable, every composition ...
1
vote
0answers
200 views

Positive and negative elements of a cyclically ordered group

I am trying to prove A property of a cyclic order on a ring. In order to do it, I need the last two properties (lemmas 1.11 and 1.12) in this question. I separated them from the original theorem ...
0
votes
1answer
35 views

Symmetric groups of sets with the same cardinality are isomorphic

Let $X$ and $Y$ be two sets s.t. $|X|=|Y|.$ Show that the groups $\operatorname{Sym}(X)$ and $\operatorname{Sym}(Y)$ of all permutations of $X$ and $Y$, respectively, are isomorphic. My attempt: ...
19
votes
3answers
4k views

Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A \stackrel{f}{\...
5
votes
2answers
28 views

Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

I'm studying for a qualifying exam in algebra, and my abstract algebra skills are quite rusty. I'm attempting to solve the following problem: Suppose that $\Phi:G\rightarrow\mathbb{Z}$ is a ...
1
vote
0answers
130 views

Apex of a cyclically ordered group

Does it make sense to introduce the new definition? Definition 2.1. An element $\pi$ of a cyclically ordered group is an apex of the group iff $\pi = - \pi \ne 0$. Considering an element $x$ of a ...
2
votes
4answers
155 views

Question in discrete mathematics about group permutations

So I have this question and i got pretty much stuck. Let $\pi$ be the permutation $$\pi= (1 2 3 4 5 6 7)\circ(1 3 5 7)\circ(2 4 6)$$ of the set $\{1,2,3,4,5,6,7\}$. Write $\pi$ as a product of ...
1
vote
0answers
120 views

A property of a cyclic order on a ring

Is this property correct? If yes, is there a better way to prove it? If not, what would be an example of a ring that does not satisfy the condition? Theorem. In a ring with non-linearly cyclically ...
0
votes
3answers
43 views

ord$(h)|\max\{\text{ord}(g)|g\in G\}$ for all $h\in G$.

Let $G$ be a finite abelian group and $n:=\max\{\text{ord}(g)|g\in G\}$. Now I have to proof that ord$(h)|n$ for all $h\in G$. My idea was: Let $g\in G$ with ord$(g)=m<n$. Then because of the ...
2
votes
2answers
30 views

In $S_{5}$ show there are $5$ elements $\rho$ with $\rho \sigma \rho^{-1}=\tau$ for given $\sigma$ and $\tau$

Let $\sigma = (12345)$ and $\tau = (13524)$, find an element $\rho$ such that $\rho \sigma \rho^{-1}=\tau$ and then show there are exactly $5$ such elements. Now I computed $\rho$ using $\rho \sigma \...
1
vote
1answer
35 views

The relationship of derived subgroup and absolute center of a group $G$

Questions: For any group $G$, the absolute center $L(G)$ of $G$ is defined as $$L(G) = \lbrace g\in G\mid \alpha(g)=g,\forall\alpha\in Aut(G) \rbrace,$$ where $Aut(G)$ denote the group of all ...
2
votes
2answers
694 views

When is the centralizer of a subgroup equal to the center?

Let $G$ be a group, and $H\leq G$ be a subgroup. When is $C_G (H)=Z(G)$? Similar to this question, which is about the centralizer of an element rather than of a subgroup: When is the centralizer and ...
0
votes
0answers
13 views

What are the closed subgroups of $p$-adic solenoid?

Let $f\colon S^1 \rightarrow S^1$ given by $f(z)=z^p$, and think $S^1 = \{z\in \mathbb{C}\colon |z|=1\}$ as a multiplicative group, so $f$ is an homomorphism. Let $S_n=S^1$ and $f_n=f$ for all $n$, ...
0
votes
0answers
33 views

From the perspective of 'group theory' , how to create high order functions? [on hold]

Say, I have a variable $a$ and $b$ I could create a thrid variable $c$ by $ x=a*b $ , or $x=f_{1}(a,b)$ In the same manner, we could have $y = f_{2}(x, a)$. We can create a lot of new variables ...
4
votes
2answers
17 views

Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
1
vote
1answer
42 views

Characterization of anti-homomorphisms

Let $G$ be a group and $G^{op}$ denotes its opposite group. It is well-known that the functor $F$ from $Grp$ to itself, defined by $$ \begin{aligned} G&\mapsto G^{op}\\ x&\mapsto x^{-1}\\ \...