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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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Are there equations which have solutions in all groups but which are not algebraicly solvable

I am not sure exactly how to phrase this problem so I appologise if it is not clear, also this is somewhat long but I wanted to explain exactly where I was with the problem. If you have any questions ...
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2answers
15 views

Subgroup conditions: inverse in the absence of an identity

This is a problem in Fraleigh's abstract algebra text. $F$ contains the set of functions from $\mathbb{R}$ to $\mathbb{R}$, and $\tilde{F}$ is the subset of functions that are non-zero at any point ...
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2answers
27 views

Presentation of abelian groups and classification.

One of my friends today asked me a question. I am unable to see a clear way to attack such a problem. Here it is: Let $G$ be a finite abelian group with generators $a,b,c,d$ and relations : $2a = 4b+...
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20 views

Cayley's theorem and relation between groups $A_n$, $B_n$, $D_n$ and $S_n$

By and large I would be happy to hear any thoughts on the facts below which from my point of view may have connections with Cayley's theorem and some relations between groups $A_n$, $B_n$, $D_n$ and $...
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12 views

How to know if an automorphism is induced by the normalizer

This is directly related to an initial question I had here. I want to followup this question with another one. Supposing I know $G\le S_n$ is a permutation group and that $C_1$ and $C_2$ are conjugacy ...
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1answer
31 views

Example of a derived series that never stabilizes.

Definition 1. Let $G$ be a group, then $[x,y]:= xyx^ {-1}y^ {-1}$, for all $x,y \in G.$ Definition 2. We define $G^{(i)}$ in the following way, $$G' = G^{(1)} = [G,G] = \{\mbox{group generated by ...
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How to check the property of projective linear group

In Isaac's Finite Group Theory Page 50, it states: A $Sylow$ $2$-subgroup $P$ of $G=PSL(2,7)$ of order 168 is contained in two maximal subgroup of $G$, each of order $24$, and $Z(P)$, which has ...
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2answers
50 views

With $G=S_{5}$ acting on $X$ by conjugation, prove $|X|=10$.

Multiple problems, that don't all fit in the title. Let $G=S_{5}$ and let $X=\{H \leq G: |H| =3 \}$. It is given that $G$ acts on $X$ by conjugation, so by mapping $(g,H)$ in $G \times X$ to $gHg^{-...
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1answer
20 views

Show a group $G$ of invertible $2 \times 2$ matrices with coefficients modulo $13$ has seven subclasses and prove it has 3 normal subgroups.

Let $G= \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \text{ with } a \text{ in } \{1,3,9\} \subset F^{*}_{13} \text{ and } b \text{ in } F_{13} \right\}$, where $F^{*}_{13}$ is the ...
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4answers
35 views

In $U_{24}$, find a smallest positive integer $n$ such that $[7]^n=[1]$

In $U_{24}$, find a smallest positive integer $n$ such that $[7]^n=[1]$ We have $24\mid 7^n-1\implies 4\mid (7^{n-1}+7^{n-2}+\dots+1)$. From intuition, $n=2$. Is there any way to find such $n$?
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1answer
23 views

Grassmannian as a quotient of orthogonal or general linear group

I'm trying to understand some different ways to construct the Grassmannian of a real vector space, but I'm having trouble getting some of the notation and definitions. One definition that I often see ...
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2answers
32 views

How to compute identity from symmetric difference? [on hold]

We know that $(P(X), \Delta) $ forms a group, where $P(X) $ is the power set on the non empty set $X$ and $A\Delta B=(A-B) \cup (B-A)$ for all $A, B\in P(X) $. Clearly $\emptyset$ is the identity ...
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3answers
56 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 ...
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2answers
55 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 $...
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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 \...
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1answer
23 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 ...
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1answer
46 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 ...
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1answer
34 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. ...
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1answer
35 views

Finding a surjective homomorphism $G \to \mathbb{Z}[\frac12, \frac13] \rtimes \mathbb{Z}$ 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] :=...
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2answers
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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 ...
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2answers
27 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 ...
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2answers
43 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 ...
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138 views

MacLane Birkhoff $\textit{ Algebra }$ vs Jacobson $\textit{ Basic Algebra I,II }$ vs Lang $\textit{ Algebra }$

I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. I've thought of the aforementioned ...
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0answers
28 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
42 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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131 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 ...
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38 views

Quotient group of matrices and permutation matrices [on hold]

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
50 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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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
147 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
96 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
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.
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0answers
39 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
26 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
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 ...
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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
156 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
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: ...
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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 \...
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2answers
53 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 ...
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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
14 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
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
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2answers
18 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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36 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
52 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 ...