The study of symmetry: groups, subgroups, homomorphisms, group actions.

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Quotient of Semi-Direct product

Let $G=H\rtimes K$, where $H\trianglelefteq G$. Suppose $N\trianglelefteq G$ and $N=A\rtimes B$, where $A\trianglelefteq H$ and $B\trianglelefteq K$. Then do we have a good description of $G/N$ such ...
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3answers
2k views

Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$

Let $G$ be a group. Show that a nonempty subset $H$ is a subgroup of $G$ if any only if $ab^{-1} \in H $ for any $a,b \in H$. The forward direction is quite easy. Suppose $H$ is a subgroup. Then by ...
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20 views

Solvability of a group

What is the intuition behind the solvable groups? It is defined by composition series. Is there any intuitive way to understand it?
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1answer
31 views

On a finite group with unique minimal subgroup

EDIT: Let $G$ be a finite group with a unique minimal subgroup. Then we know that $G$ is a $p$-group. Let $p\neq2.$ Is it true that every subgroup $H\le G$ and every quotient $G/N$ have a unique ...
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0answers
28 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
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3answers
269 views

Number of prime divisors of element orders from character table.

From wikipedia: It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group ...
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1answer
54 views

Classification of subgroup of $\mathbb{Z}^n$.

Fix an integer $n\geq 2$, can we list all subgroups of $\mathbb{Z}^n$?
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2answers
44 views

Definition of “the same”

Given a subgroup H of a group G such that $g^{-1}hg \in H$ for all $g \in G$ and all $h \in H$, I need to show that every left coset gH is the same as the right coset Hg. In the context of this ...
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46 views

Proofread my work: Expressing generators of a cyclic group

The following question comes from Serge Lang's Undergraduate Algebra(pg. 26, 3rd edition). I just learnt the concept of groups and subgroups and I spent an hour or so on tackling part (b) of this ...
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1answer
22 views

Sylow subgroups of a non abelian group $G$ with $|G|=21$ and $|G|=39$

I am trying to solve the following exercise: ¿How many Sylow subgroups has a non abelian group $G$ of order $21$ and $39$ respectively. I could do the following: a) $|G|=21=3\cdot 7$. I'll call ...
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1answer
23 views

Cycles of odd length: $\alpha^2=\beta^2 \implies \alpha=\beta$

Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2=\beta^2$, then $\alpha=\beta$. I need advice on how to approach this. I recognized that $\alpha,\beta$ are ...
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1answer
58 views

Showing associativity holds over n elements

Say we have a set $X$, with an associative binary operator $*$. How can we show that for any string $x_1 x_2 \ldots x_n$, when we insert brackets or the operation $*$, we will always get the same ...
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1answer
42 views

Normal subgroup and index problem

Let $G$ be a group and let $N$ be a normal subgroup of $G$ of finite index. Show that if $H$ is a finite subgroup of $G$ whose order is coprime with $[G:N]$, then $H$ is a subgroup of $N$. I don't ...
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1answer
22 views

Exercise on generated subgroup

Let $G$ be a finite group and $H\leq G$, $H$ cyclic. If $x \in C_{G}(H)\smallsetminus H$ then $<x,H>$ is abelian. How to prove that $<x,H>$ is also cyclic?
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12 views

Groups and U27 double check

This is just a quick question. The Group U$_{27}$=$(1,2,3,5,7,11,13,17,19,23)$ right? Or am I just very wrong here?
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3answers
191 views

Proving every element in $G$ has finite order.

Let $G$ be a group such that the intersection of all its subgroups which are different from $(e)$ is a subgroup different from $(e)$.prove that every element in $G$ has finite order. Book: Her ...
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1answer
26 views

Conditions for a finitely generated group with finite ordered generators

What are the conditions for a finitely generated group $G$ with finite ordered generators say $a_1, a_2,...,a_n$ to be finite? Note:I know that if $G$ is abelian, then it is finite. Are there any ...
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1answer
46 views

Non-isomorphic groups

How to prove that $Z/2\times Z/2$ and $ Z/4$ are not isomorphic? I think that $Z/2\times Z/2$ is not cyclic. Hence $Z/2\times Z/2$ and $ Z/4$ are not isomorphic. Thank you.
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+100

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
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1answer
51 views

Prove that if $g^{n} \in H$, n divides |g|

Let H be a subgroup of a finite group G. Suppose that g belongs to G and n is the smallest positive integer such that $g^{n} \in H$. Prove that n divides |g|. I couldn't get anywhere with this ...
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2answers
24 views

algorithm to find the order of $a$ in $(\mathbb{Z}/n\mathbb{Z})^*$ where $n$ is not prime.

algorithm to find the order of $a$ in $(\mathbb{Z}/n\mathbb{Z})^*$ where $n$ is not prime. Now I know a naive algorithm where you just keep multiplying $a$ by itself until you find it equals $1 \mod ...
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2answers
59 views

Showing if $G$ is a group then the commutator of an element is a subgroup

Given: Let $G$ be an arbitrary group, and let $a\in G$. The centralizer of $a$ is defined as $$C(a)=\{x\in G: xa=ax\}.$$ Question: Show that if $G$ is a group and $a\in G$, then $C(a)$ is a subgroup ...
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0answers
25 views

Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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2answers
310 views

Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function $$ F_G:S\times S\to\mathcal P(S)\\ F_G(a,b):=\{a\cdot b,~b\cdot a\} $$ from $S\times S$ ...
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1answer
24 views

Equivalent actions with normalizing element

Suppose $x \in G$, $G$ a group, $H$ a subgroup of G, and $G$ acts on a set $X$. Then if $x$ normalizes $H$, I know that we can say that $x$ permutes the orbits of $H$ on $X$. How can we prove that ...
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4answers
641 views

Associative law is not self evident

The statement: "It is important to understand that the associative law is not self-evident; indeed, if $a*b=a/b$ for positive numbers $a$ and $b$ then, in general, $(a*b)*c\ne a*(b*c)$." - p. 3, A. ...
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4answers
50 views

To which group is this presentation isomorphic?

$$\left\langle x_1, x_2, x_3 \,\Big\vert\, x_1^2, x_2^2, x_3^2, (x_1 x_3)^2, (x_1 x_2)^2, (x_2 x_3)^2 \right\rangle$$ is a group presentation. Could anyone tell me what does this presentation stand ...
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1answer
28 views

Index of center $Z(G)$ is finite implies the number of elements of conjugacy class is finite

Exercise Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements. I don't know how to attack the problem. I thought the ...
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15 views

(Proof-Strategy) Showing that the order of an element in the symmetric group is the least common multiple of the cycle lengths.

This question is not looking for a proof. I just want to make sure my proof-strategy makes sense and could be used to prove the following statement: Prove that the order of an element in $S_n$ ...
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11 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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1answer
26 views

Short exact sequence with binary tetrahedral group does not split

The following is a short exact sequence, where $T$ is the binary tetrahedral group (equivalently the Hurwitz units), and $Q$ is the quotient of $T$ by $\mathbb{Z}/2$. $1 \rightarrow \mathbb{Z}/2 ...
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68 views

Doubt about isomorphic groups!

Let $G$ and $H$ be two isomorphic groups.According to my present knowledge,both $G$ and $H$ have same properties and need not be distinguished.But my book says that $G$ and $H$ can behave differently ...
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1answer
19 views

Generating sets of the free group $F_k$ on $k$ generators [duplicate]

Is it true that the free group $F_k$ on $k<\infty$ generators requires at least $k$ elements to generate. I.e. does every set which generates $F_k$ have cardinality at least $k$?
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1answer
35 views

How do I construct the multiplication of a quotient group?

The question is: If $G$ is the group of all nonzero real numbers under multiplication and $N$ is the subgroup of all positive real numbers, write out $G/N$ by exhibiting the cosets of $N$ in $G$, ...
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1answer
65 views

Proving a defined group $(G,*)$ is isomorphic to $(\mathbb{R},+)$

I am studying abstract algebra and I have this question: Let $G=${$a\in\mathbb{R}|-1<a<1$} Defined an operation $*$ in $G$ with $a*b=\frac{a+b}{1+ab}$ for all $a,b \in G$ Show that $(G,*)$ ...
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1answer
31 views

Existence of finite indexed normal subgroup for a given finite indexed subgroup.

Prove that if H has finite index n then there is a normal subgroup N of G with $N \subset H $ and [G:N]$ \le $n!. I tried to solve the problem but could not done exactly. Since [G:H]=n , Let A={$a_i ...
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1answer
48 views

short exact sequences of linear algebraic groups and $K$-forms

This is probably a stupid question, but I can't figure it out. Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are ...
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496 views

Do all Groups have a representation?

I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices. Especially all elements of rotation groups can be represented by ...
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23 views

Book for practice [on hold]

Tell me books only for practice 1. Real analysis 2. Abstract algebra 3. Complex analysis I want to do only exercise For theory purpose I have so many books
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2answers
40 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
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0answers
36 views

Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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1answer
43 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
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39 views

Mapping set of integers to irrational numbers.

Mapping Integers to Irrationals..maybe even primes? Hi i'm an undergrad currently working on a research project. I recently thought of a question that I believe would help me greatly,but have ...
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39 views

Irreducible permutations in $S_n$

Let $n$ be an integer $\geq 2$ , let $\tau \in S_n$ and let $X$ be a nonempty subset of $\{1,2,\ldots,n\}$. Say that $\tau$ fixes $X$ if $\tau(X)=X$, and say that $\tau$ acts irreducibly provided the ...
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2answers
68 views

When can an infinite abelian group be embedded in the multiplicative group of a field?

This question comes from this question by user72870. That question would easily be answered if we know the cyclicity of the group in question, but, as the OP appears to be trying to prove that the ...
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1answer
35 views

Sylow p-subgroups and Sylow theorem

Find all Sylow 3-subgroups of $S_3\times S_3$? This is what I already found: Since $O(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then ...
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1answer
31 views

Nonabelian group of six elements

What is an example of a six-element group that is not abelian? I can't think of any. It is very possible that I am overthinking this. Thank you for any help.
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1answer
155 views

Why is a monoid with right identity and left inverse not necessarily a group?

This problem is from Herstein's 'Topics in Algebra'. I've thought about it a bit but haven't come up with much. Let $G$ be a non-empty set with an associative product which also satisfies: $\exists ...
2
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1answer
179 views

Show that the order of the group is at least 33

Let $G$ be a group and $a$ and $b$ be in $G$ with $a$ of order 11 and $b$ of order 3. Show that the order of $G$ is at least 33. I'm trying to do this from first principles. Obviously with lagrange, ...
3
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0answers
58 views

Can all non-archimedean groups be written as a product of archimedean groups?

We say that a partially ordered group $(G,\cdot, \geq)$ is Archimedean if for any $g,h >1\in G$ there exists some n such that $g^n > h$. All the non-archimedean groups I know of can be written ...