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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.

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21 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 ...
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2answers
30 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 ...
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3answers
95 views

Books that leaves proofs for the reader

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 ...
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0answers
32 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 =...
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1answer
41 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 ...
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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 $G'=N$.
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27 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 $N$...
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2answers
142 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 ...
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3answers
93 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?
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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 $I$ ...
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1answer
25 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.
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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 ...
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0answers
21 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 $$\...
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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 ...
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3answers
38 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 ...
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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 ...
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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 ...
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1answer
34 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: ...
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2answers
26 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 \...
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2answers
37 views

Show $N$ is a normal subgroup of $G$ where $G$ is a invertible $2 \times 2$ matrix.

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 ...
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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 ...
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0answers
12 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 $n$, ...
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0answers
32 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 ...
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2answers
16 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 ...
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0answers
12 views

Why is $A_{4}$ centerless [duplicate]

I need to prove that if $\alpha\beta=\beta\alpha$ for all $\beta\in A_{4}$, $\alpha$ must be 1. I'm thinking of finding a permutation that does not commute with any non-trivial permutation of $A_{4}$....
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35 views

Let $x\in S_{n}$ be an n-cycle. Show that $C_{S_{n}}=\langle x\rangle $. [on hold]

Let $x\in S_{n}$ be an n-cycle. Show that $C_{S_{n}}=\langle x\rangle$ .
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1answer
50 views

Subgroup of finitely-generated subgroup

Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm ...
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Subgroups of $A_n$ which contain a normal subgroup which order divisible by $n$ [on hold]

What are those subgroups of the alternating group $A_n$ which contain a transitive normal subgroup on the $n$ letters?
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1answer
46 views

Abelianization of $\mathbb{Z}\ltimes_\varphi \mathbb{Z}^n$

i would like to ask how to compute the abelianization of the semidirect product $\mathbb{Z}\ltimes_\varphi\mathbb{Z}^n$ where the action is $\varphi(k)v=A^k v$ where $A$ is a fixed invertible matrix ...
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2answers
29 views

Proving that order of a cyclic group $G$ is divisible by a number

Suppose that $\exists x,s.t. |x|=20$ and $\exists y,s.t. |y|=16$ and $x,y \in G$ where $G$ is a cyclic group. How can I show that order of $G$ is divisible by 80? I was thinking of using the fact ...
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1answer
31 views

Group of the Image of a character of a finite abelian group is cyclic [on hold]

Let $G$ be a finite abelian group and $\hat{G}$ be its dual group. $\gamma$ is a character of $G$, i.e. $\gamma:G\rightarrow \mathbb{C}$, $|\gamma(x)|=1$, $\forall x \in G$ and $\gamma(x+y)=\gamma(x)\...
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0answers
40 views

If $x∊S_n$ normalizes but does not centralize a subgroup of prime order $P$, show that $x$ fixes at most one point in each orbit of P$. [on hold]

Let $P⊆S_n$ be a subgroup of prime order and suppose $x∊S_n$ normalizes but does not centralize $P$. Show that $x$ fixes at most one point in each orbit of $P$.
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1answer
55 views

Representation theory of SO(p,q)

For a long time now, I have tried to look for the representation theory of $SO(p,q)$. I am in particular interested in the unitary irreducible representations and the bilinear Hermitian form on the ...
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0answers
37 views

Proper Set Notation

I am currently working on a group theory project, and have come to the topic of conjugacy classes. For a group $\zeta$, the conjugacy class containing the element $A$ is defined to be the set of all ...
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2answers
31 views

Why $f^{'}_{*}$ group homomorphism exist in 'only if' part of the lifting criterion proof?

In Hatcher, the lifting criterion states (Prop 1.33): Suppose given a covering space $p: (X^{'},x^{'}) \rightarrow (X,x_0)$ and a map $f: (Y,y_0) \rightarrow (X,x_0)$ with $Y$ path-connected and ...
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2answers
23 views

Why there exists a group homomorphism $f_t$ such that $f = pf_t$?

Let $p: X^{'} \rightarrow X$ and $f: Y \rightarrow X$ be two group homomorphism such that $f(Y) \subset p(X^{'})$. Why does there exists a homomorphism $f_t:Y \rightarrow X^{'}$ such that $f = pf_t$? ...
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0answers
30 views

Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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1answer
37 views

Prove that if $H \leq G$ has exactly $k$ conjugates in $G$ then there exists a homomoprihsm $\alpha: G \rightarrow S_k$ such that…

Prove that if $H \leq G$ has exactly $k$ conjugates in $G$ then there exists a homomoprihsm $\alpha: G \rightarrow S_k$ such that $k$ divides $im(\alpha)$ Proof: Let $G$ act on the set of conjugates ...
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1answer
24 views

Group Theory - Correspondence Theorem [on hold]

Q. Show that $K=\{1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)\}$ is a normal subgroup of $S_4$, and find all subgroups of $S_4$ containing $K$ by using Correspondence Theorem.
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2answers
29 views

$S_3$ is soluable but not nilpotent

Claim: $S_3$ is soluable but not nilpotent Proof: $S_3 \cong D_6$. $D_6$ has a subgroup of order 3 generated by the rotations, $\langle r\rangle$. $|\frac{D_6}{\langle r\rangle}|=2$ so $\langle r\...
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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 ...
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1answer
25 views

Cayley's theorem: Is $C_5$ isomorphic to $ \langle (1 2 3 4 5) \rangle \leq S_{5}$?

I want to use Cayley's theorem to determine a subgroup in $S_n$ ( for n as small as possible) which is isomorphic to $C_{5}$. I believe this subgroup to be $ \langle (1 2 3 4 5) \rangle $. Here is ...
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0answers
48 views

Double centralizer of derived subgroup

Let $G$ be (finite) matabelian group; define $W(G):=C_G(C_G(G'))$; $G'$ is derived subgroup of $G$. If $\mathcal{F}$ is the collection of all maximal abelian normal subgroups of $G$ which contain $G'...
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1answer
60 views

Why is the Orbit Stabiliser Theorem intuitively true?

Why is the OST intuitively true? (Specially for the finite groups but also infinite groups) I understand the proof and the steps, but it is not obvious to me like let’s say Intermediate Value theorem.
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1answer
28 views

If $G$ is nilpotent of class 2 and if $a \in G$, then the function $\alpha: G \rightarrow G$ …

If $G$ is nilpotent of class 2 and if $a \in G$, then the function $\alpha: G \rightarrow G$ defined by $x \rightarrow [a,x]$ is a homomorphism. Proof: Let $x,y \in G$ Then $\alpha(xy) = [a,xy] = [a,...
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1answer
35 views

Regarding the Intersection of $p$-Sylow Subgroups

The Problem: Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Prove that a normal $p$-subgroup of $G$ is contained in every $p$-Sylow subgroup of $G$. My Attempt(s): Here's a slight ...
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3answers
100 views

“Distinguished element” in group theory [on hold]

I have asked that given a short list of triples of a set, a binary operation on the set, and a distinguished element, whether they forms a group. For example, the set is Z, the operation is addition ...
1
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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}\\ \...
2
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1answer
37 views

Normalizer in matrix groups

I have the problem of calculating the normalizer of $\begin{bmatrix}      \lambda & 0 \\ 0 & \lambda^{- 1} \end{bmatrix} $ in the group $\begin{bmatrix}      \cos (\theta) & -\sin (\theta) ...
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1answer
54 views

How to understand the structure of the interesting graph obtained from the group?

Let $G = A_5$ and $H < G$ is subgroup of $A_5$ generated by $(12)(34)$, and $(125)$. Define graph $\Gamma$ by a vertex set is element of $G$ and elements $x$ and $y$ adjacent if $|H^x \cap H^y| =...