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.

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Solvability of word problem in group

I am fairly new to abstract algebra. So, I apologize if my question is too trivial. I am trying to prove that $G=\langle x,y \mid xy=yx; x^2=1\rangle$ has a solvable word problem. My idea is to show ...
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Question on subgroups of reductive groups

A linear algebraic group $G$ over some field $k$, which I assume being of characteristic 0, is reductive if $R_u(G^0_{\overline{k}})$ is trivial, where $R_u$ denotes the unipotent radical, $G^0$ is ...
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224 views

Proving that two quotient groups are isomorphic

Given a group isomorphism $\phi:G\rightarrow H$, and a subgroup $K \subset G$, I need to show that there is a group isomorphism $G/K\cong H/\phi(K)$, where $\phi(K)=\{h\in H \,\,\text{such ...
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What is known about automorphism group cardinality?

What is known about automorphism group in general and about $|\text{Aut}(G)|$? Is it true that $|\text{Aut}(G)| \le |G|$? Exist any algorithm to build $\text{Aut}(G)$ for given $G$? $G$ is finite.
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261 views

Reference for proof of Kaloujnine-Krasner

The theorem of Kaloujnine-Krasner says Given two groups $D$ and $Q$, the wreath product $D \wr Q$ contains an isomorphic copy of every extension of $D$ by $Q$. I am looking for an English ...
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Concerning Cyclic Groups

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution). Here it is: ...
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138 views

How to find [G:H]?

Let $F$$=GF(11)$ be finite field of 11 elements. G is group of all non-singular n$\times$n matrices over F.$H$ is subgroup of those matrices whose determinant is 1. Then $[G:H]$=?
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143 views

Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
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132 views

If $G/Z(G)$ is abelian, does $G$ have to be abelian? [closed]

If $G/Z(G)$ is abelian, does $G$ have to be abelian ? $Z(G)$ is the centralizer of G, i.e. $$Z(G)=\{h \in G \ \ | \ \ hg=gh \forall \ \ \ g\in G\}$$ Thanks for your help in advance
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111 views

Is $\mathbb{Z}_7^*$ cyclic?

Determine whether the following sentence is correct or not. $$ \mathbb{Z}_7^* \text{ is cyclic. }$$ Is $\mathbb{Z}_7^*$ the same as $\mathbb{Z}$ without $0$?? If it is ...
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33 views

Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
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40 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
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1answer
60 views

An interesting puzzle for some, confusing for me

Suppose that $a$ is of odd order $k$ and $bab=a$. I need to show that $b$, must be of order $2$. We can prove this anyway we want to, but our hint is to expand $(bab)^k$ and re-associate and then ...
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46 views

Groups - identity - abstract algebra

Let $G$ be a finite cyclic group with $n$ elements, $$G = \{a_1 , a_2,\dots,a_n\}$$ Let $x = a_1 \cdot a_2 \cdot \dots \cdot a_n$ 1) Show that $x^2 = e$, where e is the identity of group $G$. 2) If ...
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49 views

Find the order of the elements in the given groups

I have to find the order of the following elements in the given groups: $(1 \ \ 2 \ \ 3) \ (1 \ \ 2\ \ 4) \text{ in } S_5$ $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 1 ...
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1answer
68 views

classifying space of permutation groups

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?
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47 views

Normal implies characterstic

We know that if in a group G , HcharG then H is normal in G But converse is not true. converse is true when (|H|,|G/H|)=1. Then how we show this. Please help me. Thanks.
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Can we conclude from that, that there is an homomorphism between the group $G$ and the group $(\mathbb{Z}_3,+)$?

We have the third order group $G=\{1,g,x\}$, whose operation is the multiplication. To calculate the multiplication table we do the following: $1 \cdot 1=1, \ \ \ 1 \cdot g=g, \ \ \ 1 \cdot x=x$ ...
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250 views

Kernel of homomorphism between two cyclic groups of diferent order

In Malik's abstract algebra one can find the following exercise (and I paraphrase): Let $f$ be a homomorphism from a cyclic group of order 8 onto a cyclic group of order 4. Determine $\ker f$. ...
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4answers
642 views

Show that an abelian group $G$ of order 55 must be cyclic.

I know that in order to be cyclic: A group G is called cyclic if there exists an element g in G such that G = ⟨g⟩ = { $g^n$ | n is an integer } by wikipedia. But I just get lost in how simple it looks ...
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259 views

$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$

I 've just read from An introduction to the theory of groups from Rotman's the following theorem: Theorem 2.19. Let $p$ be a prime. A group $G$ of order $p^n$ is cyclic if and only if it is an ...
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64 views

Free groups of rank greater than 2

I'm trying show that a free group of rank $\ge2$ is non abelian, but I have no idea to prove this. Any suggestions?
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255 views

Techniques for disproving group isomorphism

Suppose I wanted to find out if $f:\mathbb{Z}_6\rightarrow S_3$ is an isomorphism. Clearly, $f$ is bijective. It remains to show that $f(a+b)=f(a)\circ f(b)\;\forall a,b\in \mathbb{Z}_6$. For this ...
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65 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
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59 views

$C\subseteq D$. Prove $\mathcal P (C)$ is a subgroup of $\mathcal P(D)$.

Let $C$ and $D$ be sets, with $C\subseteq D$. Prove $\mathcal P(C)$ is a subgroup of $\mathcal P(D)$. I can't easily see a proof for this, so I tried working on a counterexample. If I could just ...
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25 views

Show that the groups are homomorphic

T={the n-th roots of unity} is a cyclic group of order n with the multiplication as operation. How can I show that there is a group homomorphism between this group and $(\mathbb{Z}_2,+)$ ?? Do I have ...
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56 views

A question in definition of group rings

In definition of a group ring $RG$ with elements $∑f_g g$ (where $g\in G$ and $f_g\in R$), are we supposed that $f_g$'s commute with $g$'s? I mean could we identify the above formal summation with $∑ ...
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1answer
38 views

SL(2,5) and SL(2,11)

there is a problem in my textbook as follows: Why the finite group $SL(2,5)$ is isomorphic to a subgroup of $SL(2,11)$? Thanks for the answers
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79 views

Showing that no homomorphism $ϕ:∏ _{i∈N} \Bbb Z→\Bbb Z$ can send all $e_i$ to 1

why is that no homomorphism $\phi:\prod_{i\in\mathbb{N}}\mathbb{Z}\to\mathbb{Z}$ can send all $e_i$ to 1? In fact I saw a proof in MO using 2-adic integers...but I know very little about those topics. ...
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682 views

Homomorphism between symmetric group and general linear group of order n. [closed]

I am having trouble proving the following: Show that $f: S_n \to GL_n(\mathbb{R}),\;\: f(x)=A_x$ is a homomorphism where $A_x$ is the permutation matrix associated with $x$. $S_n$ is the ...
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104 views

Cyclic group 60

In a cyclic group of order 60 find the elements of order 12. then find the number of element that satisfy $x^{12}=e$ So if $x^3=e$ then $x^{12}=e$ And I know $x=e$. what next do I do? Finally find ...
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1answer
87 views

Lang's Representation of Third Isomorphism Theorem

I wanted to ask what is the intuition behind the commutative diagram representation of the third isomorphism theorem in Lang's Algebra, page 17. Namely, the one that states $$(G/K)/(H/K)\approx G/H$$ ...
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21 views

Show that $H:=\{m/2^n:m \in \mathbb{Z}, n\in \mathbb{N}\}/\mathbb{Z}$ doesn't have a proper maximal subgroup.

Show that every proper subgroup of $H:=\{m/2^n:m \in \mathbb{Z}, n\in \mathbb{N}\}/\mathbb{Z}$ is contained in a larger proper subgroup of H. I've no idea. I wanted to prove by contradiction, for ...
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24 views

Question about sum of abelian groups.

So there's a statement in Lang that I would like to understand better. It's contained in his proof of the following statement: Every finite abelian p-group is isomorphic to a product of cyclic ...
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The simply connected form of a semisimple algebraic group

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, so that $G$ is an almost-direct product of its minimal closed connected normal subgroups of positive dimension, ...
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Cyclic group of order 8

In a cyclic group of order 8 show that element has a cube root. So for some $a\in G$ there is an element $x \in G$ with $x^3=a.$ Also show in general that if $g=<a>$ is a cyclic group of order ...
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Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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39 views

How to show homomorphism of groups

How can I show that $G=\{ 1, g \} \cong (\mathbb{Z}_2,+)$ ? Do I have to find a function $f:G \to \mathbb{Z}_2$ such that: $$f(a \cdot b)=f(a)+f(b)$$ where $a,b \in G$? If so, how can I find such ...
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67 views

Suppose G is a group which has only one element a such that |a| = 2. Prove that xa = ax, for all x ∈ G.

I know the following are true. 1) There is an inverse of a 2) There is an identity element (e*a) = a In this case, e = 1 and the inverse of a is 1/|2|. However, if a is the only element in G and a ...
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1answer
51 views

Equality of subgroups of finite cyclic groups

My abstract algebra text has the following proof of the statement, "Let $G$ be a cyclic group with $n$ elements and generated by $a.$ Then $\langle a^s \rangle = \langle a^t \rangle \iff \gcd(s,n) = ...
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Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group ...
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Explain why every proper subgroup must be cyclic.

Let G be the group with presentation $< x, y : x^4=1; x^2=y^2; xy=yx^{-1} >$. Find the order of each of the eight elements. How many elements of order 2 are there? Explain why every proper ...
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62 views

Group which is not cyclic

We know that the group $U(2^n)$ is not cyclic for $n \geq 3$. But I want to prove that $U(n^2-1)$ is not cyclic for $n>2$. Please help me
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Supersolvable groups of order $pq^m$

According to Burnside's classification in his book "Theory of groups of finite order", one of the types of non-abelian groups of order $pq^2$ ($p$ and $q$ are distinct primes), has the presentation ...
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List of elements in the group

There are $8$ elements in the group $\mathbb{Z_4}\times \mathbb{Z_4^*}$. I am having trouble listing the eight elements. Note: $\mathbb{Z_4^*}$ means group of units in $\mathbb{Z_4}$.
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139 views

Subgroup generated by and element

Consider the group G = {1, 3, 5, 7} under multiplication modulo 8. What is the order of the element 5? I know that the order of an element is the order of the subgroup generated by the element. so ...
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104 views

Show that there are infinitely many primes $p$ such that $p = 1 (\mod q)$ in a very specific way

I friend of mine has shown the following: Let $n \in \mathbb{N}$, and $q$ an odd prime number. Any $p$ dividing $1 + n + \cdots + n^{q-1}$ satisfies $p \equiv 1 (\mod q)$, whenever $ n \not \equiv ...
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Constructing a certain sort of short exact sequence

If I have a group $N$ and an automorphism $\phi$ of it such that $\phi^n$ is inner (suppose by conjugation by $x \in N$) and $\phi(x)=x$, how can I prove that there is a group $G$ that fits into the ...
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39 views

self-normalizing subgroups of the unitary groups

I have an open question: what is known about self-normalizing subgroups of the unitary groups? By "self-normalizing subgroup," I mean a subgroup that is its own normalizer in the larger group. Has ...
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86 views

Alternative to the Frattini argument

If $G$ is a finite group with $H \trianglelefteq G$, and $P$ is a Sylow $p$-subgroup of $H$, then we can show that $G = N_G(P)H$. While I'm now aware of the (admittedly much simpler/nicer) Frattini ...