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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366 views

Proof for: semidirect product of solvable groups is solvable

Do you know the proof for the following statement or where I can find it? Semidirect product of solvable groups is solvable? I thought that this property is so ...
3
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3answers
219 views

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$? [duplicate]

Let $ (G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
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1answer
37 views

Simple subgroup of transitive group

Suppose $Q \in Syl_n(G)$ and $Q$ is not normal in $G$, $Q$ is generated by an element of order $n$, where $G \leqslant S_n$ for prime $n$ and $G$ acts transitively on $\{1,...,n\}$. Define H to be a ...
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1answer
57 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
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1answer
474 views

The definition the group of rigid motions in $\mathbb R^3$ of a tetrahedron

In Dummit & Foote's Abstract Algebra text, page 28 the following problem appears: 9. Let $G$ be the group of rigid motions in $\mathbb R^3$ of a tetrahedron. Show that $|G|=12$. Apparently, ...
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102 views

Is there a name for this object? (Like a group, but the inverse is not necessarily a member of the set)

A group is a set $G$, together with a binary operation $\cdot$ that is closed - if $f\in G$ and $g \in G$ then $f\cdot g \in G$ is associative - $(f \cdot g) \cdot h = f \cdot (g \cdot h)$ has an ...
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1answer
29 views

Symmetry group of a non abelian group manifold

Reading an article ...$B$ is itself the manifold of some group $H$. It should be noted that, if $H$ is a non abelian group, the symmetry group $G$ of the group manifold is not $H$ but ...
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2answers
271 views

When is the centralizer equal to the normalizer?

I have a very basic understanding so correct me if I'm wrong. If the group is abelian, then ever element in that group commutes with a set as well as every element in that set. So would that be the ...
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2answers
33 views

Show that $H_s$ is subgroup of $G$

Let $S$ be any subset of a group $G$. Show that $H_s = \{x \in G\mid xs = sx \text{ for all } s \in S\}$ is a subgroup of $G$.
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67 views

An “Arbitrary Free Product of Groups”

I've seen written the following sentence: Let $G_i$ be a collection of more than one non-trivial group. Prove that their free product is non-abelian. Now if $\{G_i\}$ denotes the collection ...
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1answer
28 views

Factor group computations

Continuing my independent study of Fraleigh (See my other posts which the users of this forum have been very helpful in answering). Section 15 on Factor Group Computations. I'm confused about ...
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86 views

Finding another representation of a group represented by a set of matrices

I want to find the representation of the group $$ \left\{ \left. \left( \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ c & 0 & ad & 0 \\ 0 & ...
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2answers
203 views

Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
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3answers
53 views

Groups statements equivalence

Problem statement Let $G$ be a group and $H \subset G$ a subset. Prove that the following statements are equivalent (i) $H$ is a subgroup (ii) $H$ is nonempty and for any $x, y \in H, xy^{-1} \in ...
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1answer
59 views

About product of $k$ commutators

I need help in folloving lemma. Lemma: Any product of $k$ commutators is expressible in the form $a_1^{-1}a_2^{-1}...a_{2k}^{-1}a_1a_2...a_{2k}$ I guess author means that any product ...
3
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1answer
41 views

Specify finite group satisfying two conditions

Let $G = \left\{ 0, 1, 2, 3, 4, 5, 6, 7\right\}$ be a group with the operation $\circ$, which satisfies the following conditions: \begin{align} a \circ b \leq a + b \quad & \forall a,b \in G & ...
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2answers
51 views

Find the nontrivial proper subgroups of $\Bbb Z_{2}\times \Bbb Z_{2}$.

All i know is that the elements of $\Bbb Z_{2}$ are $\{ 0, 1 \} $. Could anyone please explain how to compute the nontrivial proper subgroups of $\Bbb Z_{2} \times \Bbb Z_{2}$. It would be very ...
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0answers
45 views

For a cyclic Sylow p subgroup of a group G, when is $C_{G}(P)$ = $P$?

Suppose $P$ is a Sylow p subgroup of a group $G$ and that $P$ is cyclic and P is not a normal subgroup of G. Does $C_{G}(P)$ = $P$? Clearly $P$ is contained in its centralizer since $P$ is abelian. ...
2
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1answer
50 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
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1answer
65 views

Group Theory commutative diagram

Im in the following situation: Let $G$ be a finitely generated group. $ \mathbb{Z}^n \rightarrow \mathbb{Z}^n\times G \rightarrow G$ be a split exact sequence. And let $X\rightarrow Y \rightarrow ...
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91 views

Prove that G is cyclic and has exactly two nontrivial subgroups.

1- Let G be an abelian group of order 2p, where p is an odd prime. Prove that G is cyclic and has exactly two nontrivial subgroups. Write out in full each theorem or other previously proven result ...
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1answer
424 views

If a finite group $|G|$ acts transitively on a set $X$ with $|X|=2^n$, $n \geq 1$, then $G$ has an involution with no fixed points

Let $G$ be a finite group acting transitively on a set $X$, where $|X| = 2^n$ for some $n \geq 1$. Show that some element of $G$ acts as an involution with no fixed points. While it is fairly easy ...
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1answer
120 views

Groups of order $4p^2$

I am interested about a Fact (if it is right) of the structure of groups of order $4p^2$. Let $G$ be a nonabelian groups of order $4p^2$, classify all this groups.
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89 views

Proving that the order of $(a,b)$ is the lcm of $|a|,|b|$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Prove that the elements $(a,1)$ and $(1,b)$ of $A \times B$ commute and deduce that the order of $(a,b)$ is ...
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1answer
101 views

How to show that group of order 760 is not simple? [closed]

How to show that group of order 760 is not simple? By Sylow's theorem $n_{19}=20$, and $o(N(P)) = 38$, but how to continue after this? Thanks for any help
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79 views

normal subgroups and factor group

This is my question: Let $G$ be a finite group and $H\triangleleft G$ a normal subgroup. Prove that $|G/H| =|G|$ if and only if $H = \{e\}$. And this is my solution: first we need to show that the ...
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25 views

every k cyclic is a product of at least k-1 distinct tranpositions

There is a theorem says if $A$ in $S_n$ is a $k$ cycle, and $A = a_1 a_2 a_3 \dots a_m$, where $a_i$ are transpositions, then $m \geq k-1$. But how to show there are at least $k-1$ distinct ...
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2answers
115 views

Ambiguity of Quotient Groups

Abstract algebra's "equality vs. isomorphism" problem rears its confusing head once again. I was thinking about the identity $(G \times H)/H \cong G$. It's pretty easy to justify by stating that $H ...
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597 views

Groups of order 12 that aren't isomorphic [duplicate]

Give examples of four groups of order 12 no two of which are isomorphic. So far I've thought of $Z_{12}$ and $D_6$. Thanks!
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1answer
71 views

Terminology concerning conjugation in groups of functions.

If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other. If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by ...
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1answer
63 views

Unique intermediate subgroup and double coset relation II

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is there $\alpha \ge 1$ such that if ...
6
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3answers
622 views

How is the quotient group related to the direct product group?

I'm trying to understand what the relation is between the direct product and the quotient group. If we let $H$ be a normal subgroup of a group $G$, then it is not too difficult to show that the set ...
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1answer
87 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields ...
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2answers
181 views

Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
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54 views

Groups and subgroups

I have been told that {0, 2, 4, 6, 8} is a subgroup of the multiplicative integer mod 10. I know that the operation is multiplication, so I understand that every element has its inverse within the ...
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1answer
82 views

The choice of generators and the order of $\operatorname{Aut}(G)$.

A group homomorphism $G \to G$ is fixed by its values on a set of generators of $G$. Consider the order of the automorphism group $\operatorname{Aut}(G)$. Let $S = \{s_1,s_2,\ldots,s_n\}$ be a ...
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1answer
51 views

Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
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1answer
68 views

Finite cyclic group

Can anyone give me a specific example of this: Let $G=\langle a\rangle$ be a finite cyclic group of order n. If $m\in \mathbb{Z}$, then $\langle a^m\rangle =\langle a^d\rangle$, where $d=\gcd(m,n)$ ...
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1answer
416 views

Quotient of the real numbers over the rational numbers

What is the quotient group of the real numbers over the rational numbers seen as a quotient of groups? Is there a common way to represent the quotient $\frac{\mathbb{R}}{\mathbb{Q}}$?
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2answers
237 views

Interpretation of Second isomorphism theorem

I have a question about the Second Isomorphism Theorem.(Actually my book called it the first), namely, let $G$ be a group, $N$ is a normal subgroup of $G$, and let $H$ be any subgroup of $G$, then $ ...
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0answers
51 views

Interpretation of the complement $S_5 \setminus D_6'$.

On account of a homework question $$ \text{does there exist an injective homomorphism } D_6 \to S_5 \text{ ?}$$ which I solved positively, I raised the question $$ \text{what does the complement ...
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68 views

How can I prove that $C_{S_4}((12)(34))$ is a subgroup of order 8?

I would like to know if there is a smart method (i.e. avoid to make all by hand) to understand the order of $C_{S_4}((12)(34))$. The only thing I thought is that ...
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2answers
138 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
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1answer
149 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
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2answers
113 views

Dihedral group, semi direct product, and Klein 4 subgroup

Can someone help me out my confusion? If the dihedral group $D_8$ (of order $8$) is the semi-direct product of $\mathbb{Z}_4$ and $\mathbb{Z}_2$, how is it possible that it contains the non-cyclic ...
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2answers
264 views

Group Theory - Quotient group notation?

What is the difference between the following terms: $\mathbb{Z}_{4}$ , $\mathbb{Z}/4$ and $\mathbb{Z}/{4}\mathbb{Z}$ ? I am pretty sure the first one is the cyclic group with addition modulo 4... ...
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1answer
61 views

Notion of group generators

I'm asking myself what the meaning of a statement like the following is: Let $G$ be a group and $T_i \subseteq G$ be a family of subgroups of $G$ indexed by a possibly infinite set $I$. Now let ...
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157 views

Prove $S_4$ has only 1 subgroup of order 12

The subgroup in $S_4$ that I know has order 12 is the subgroup of all even permutations, otherwise known as the alternating group $A_4$. However, I know this from a fact and not because I am able to ...
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40 views

Subgroup of $S_4$ with order 12 [duplicate]

I need to find a subgroup of $S_4$ that has order 12 other than the subgroup of even permutations or anything isomorphic to it. What is the procedure to go about finding such a subgroup?
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1answer
182 views

Show that the p-Sylow subgroup is normal in $G$

Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x) = x$ implies that $x = e$. Prove that for ...