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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Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
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Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
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88 views

Number of conjugacy classes in a finite non-abelian simple group

Can we say that every finite non-abelian simple group has at least 4 non-identity conjugacy classes?
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40 views

Separating invariant of a group action

Let $G = (\mathbb{R},+)$ be a group, $M = \mathbb{R}^2$, $$\omega \colon \mathbb{R}\times\mathbb{R}^2\to \mathbb{R}^2, \quad \left(t, (x,y)\right) \mapsto(x+t,y-2t)$$ and $$\iota \colon\mathbb{R}^2 \...
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503 views

Operations on power set

Which kind of operations on a power set leads to a group/monoid? Known to me are: - intersection - union - symmetric difference - complex product of a group Ate there some more examples?
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360 views

Isomorphism classes of abelian groups of certain order

working on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space \...
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Isomorphism group between $\mathbb R$ and one of it's propre subgroups.

Is there a subgroup $H$ of $(\mathbb R,+)$ such that $H \neq \mathbb R$ and $H \simeq \mathbb R$ as groups?
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173 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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52 views

Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
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121 views

Are all torsion groups finite groups?

Are all torsion groups finite groups? I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?
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162 views

How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of $\...
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88 views

What does the notation $H=\{ a | a^2=e \}$ mean?

What is the meaning of notation $H=\{ a | a^2=e \}$? Is it the same as $H=\{a,a^2=e\}$? (Here $a$ is an element of some group, with identity $e$.)
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95 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
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85 views

On infinite groups with unique minimal subgroup

Let $\operatorname{Sub}(G)$ be the lattice of all subgroups of an infinite (abelian) group $G$. If $\operatorname{Sub}(G)\setminus\{\{1\}\}$ has a minimum element which is a cyclic subgroup of prime ...
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83 views

Presentation of a group: Show that $\langle a|a^2\rangle =\{1,a\}$.

Example: $\langle a|a^2\rangle=\{1,a\}$. After reading the definition of presentation of a group, I find myself cannot understand the above example given. I don't know which part of the definition I ...
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50 views

When can a group be decomposed into a direct product of smaller groups?

Is there any general condition that a group must satisfy in order to be decomposable into a direct sum or product of smaller groups? And what happens if one replaces 'direct product' with semi-direct ...
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85 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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63 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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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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183 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
64 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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86 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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85 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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20 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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100 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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75 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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157 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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68 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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Showing if $G$ is a group then the centralizer 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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57 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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82 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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36 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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86 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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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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80 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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35 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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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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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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28 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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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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97 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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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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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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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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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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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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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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Why is a monoid with right identity and left inverse not necessarily a group? [duplicate]

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 ...
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186 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, ...
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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 $(G,*)$ ...