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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$\langle x^n\rangle \cap \langle x^m\rangle = \langle x^{ lcm(m,n)}\rangle$ [duplicate]

Possible Duplicate: Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$ If $G=\langle x\rangle$ is a finite cyclic group that $\langle x^n\rangle \cap \langle x^m\rangle = ...
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42 views

About rotations of sets of vertices of a regular $p$-gon.

This is something I've been thinking about lately, and I don't seem to understand the problem well enough. There is a motivation to this problem, but I don't think giving it would be productive since ...
3
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1answer
95 views

Formulation of holomorph in abstract algebra

Let $H$ be a group of order $n$, $K$ = Aut($H$), and $G$ = $H \rtimes K$ where $\rtimes$ is the semidirect product with respect to the identity homomorphism $\varphi$. Let $G$ act on the set $X$ of ...
-1
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3answers
744 views

Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$ [duplicate]

This group theory problem has stumped me. I want to prove that if $G=(x)$ is a finite cyclic group that $(x^n) \cap (x^m) = (x^{\operatorname{lcm}(m,n)})$ for all integers $m$ and $n$, where $(x)$ is ...
15
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3answers
2k views

Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers

Let $\hat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$. Since $\hat{\mathbb{Z}}$ is the inverse limit of the rings $\mathbb{Z}/n\mathbb{Z}$, it's a subgroup of $\prod_n ...
1
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2answers
180 views

No group can have exactly 3 elements of order 3 (T/F)?

True of False? This is a question from a past exam that I'm practicing on. Thanks.
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3answers
158 views

Can you make noncommutative groups of order $24$ from these groups?

I do know of $3$ classes of groups (up to isomorphism) of order $24$ that are commutative (direct products of $\mathbb{Z}$/(factors of $24$)$\mathbb{Z}$. Can you just take the semi direct product ...
4
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2answers
86 views

Question about a step in the proof of the second isomorphism theorem

My book says: Since $B$ is normal, the quotient group $AB/B$ is well defined. I want to know why the "well-definedness" of $AB/B$ is dependent on $B$ being normal. Let me know if you need more context ...
7
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1answer
105 views

Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
2
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3answers
108 views

Co-Existence of the Primary Decomposition Theorem with the Fundamental Theorem of Finitely Generated Groups

Let $A$ be an abelian group of order $n = p_1^{\alpha_1} \cdot \ldots \cdot p_k^{\alpha_k}$ (i.e., $n$'s unique prime factorization). The Primary Decomposition Theorem states that $A \cong ...
2
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2answers
305 views

An epimorphism in $\text{Grp}$ without right inverse?

Exercise 8.24 in Aluffi's Algebra: Chapter 0 asks us to find an epimorphism in $\text{Grp}$ without right inverses. I happen to know that epimorphisms in $\text{Grp}$ are surjective, so we need a ...
1
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2answers
117 views

Computing the order and cyclicity of quotients of direct products.

Determine the order of $(\mathbb{Z} \times \mathbb{Z} )/ \langle(2,2)\rangle$ and $(\mathbb{Z} \times \mathbb{Z} )/ \langle(4,2)\rangle$. Are the groups cyclic? I've read many solutions on the ...
3
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1answer
113 views

Motivation for the term “transitive” group action

I have two questions: In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
8
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3answers
591 views

Abelian Group Element Orders

I want to show that if a finite abelian group has elements of order $m$ and $n$ then it will have an element of order $\text{lcm}(m,n)$. First I proved the lemma if $a$ has order $m$ and $b$ has ...
0
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1answer
295 views

Factor group of quaternion group

I have $Q_8=\{I,A,A^2,A^3,B,AB,A^2B,A^3B\}$ where $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$ Now I take a look at $Q_8/N$ ...
0
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1answer
55 views

Central series of $D_4$

I am trying to write down the central series of $D_4$. i.e $e$=$Z_0$ $\subset Z_1 \subset Z_2........\subset Z_l=D_4$ , where $Z_1=Z(D_4)$ and $Z_{i+1} $ is defined such that $Z_{i+1}/Z_i=Z(G/Z_i)$, ...
0
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1answer
175 views

Does an Internal Direct Product $G = N_1 \cdot \ldots \cdot N_n$ imply the $N_i$ are Normal with Trivial Intersection?

If we let $G$ be a group with $n$ subgroups $N_i$ such that $\prod_{i=1}^n N_i= G$ $N_i \cap N_j = \{e\}$ for all $i \ne j$ s.t. $1 \le i < j \le n$ $N_i \unlhd G$ for any $1 \le i \le n$ then ...
0
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0answers
92 views

About Sylow systems

I need some help proving this proposition. "$G$ is a solvable group if and only if $G$ has a Sylow system" (Sylow system: a set $S$ of Sylow subgroups of $G$, one for each prime dividing $|G|$, so ...
7
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3answers
922 views

How can I prove that every group of $N = 255$ elements is commutative?

There was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_17 = ...
6
votes
1answer
208 views

An infinite $p$-group may not be nilpotent

It is well-known fact that every finite $p$-group $G$ is nilpotent. I am asking to have a counter example when $G$ is infinite $p$-group. Thanks.
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5answers
101 views

According to one of the Sylow theorems, if $n_p = 1$, then the p-Sylow subgroup is normal. Why?

I've used this fact several times, and its about time I knew why its true.
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2answers
3k views

Are normal subgroups transitive?

Suppose $G$ is a group and $K\lhd H\lhd G$ are normal subgroups of $G$. Is $K$ a normal subgroup of $G$, i.e. $K\lhd G$? If not, what extra conditions on $G$ or $H$ make this possible? Applying the ...
1
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1answer
280 views

Given elements $x,y$ of a group $G$, where both $x,y$ have finite order, what is the order of $x*y$?

I've read somewhere that if $x,y$ commute, and $gcd(|x|,|y|) = 1$, then $|x*y|$ is the product of their individual order, but I don't even know why the criterion of commutativity is needed there. ...
1
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1answer
98 views

Prove that for $n \geq 5$, $A_n$ is the only subgroup of $S_n$ such that $|S_n/G| < n$

This is an exercise in Dummit and Foote (4.6.3) I'm doing for revision: prove that for $n \geq 5$, $A_n$ is the only proper subgroup of $S_n$ such that $|S_n/G| < n$. ($A_n$ is the alternating ...
2
votes
1answer
49 views

What is the orbit space generated by this action?

Let $G$ be the group generated by the two transformations $f_1,f_2:\mathbb{R}^2\to \mathbb{R}^2$ given by $f_1(x,y)=(x+1,y)$ and $f_2(x,y)=(x+1,-y)$. What is the orbit space generated by ...
0
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1answer
148 views

Possible mistake in Boothby's Manifolds book

This question is from section 7, chapter 3 of An introduction to Differentiable Manifolds By William M. Boothby. $G_{x_0}$ is the stabilizer of $x_0$. The relation that gives sense to the quotient ...
2
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2answers
210 views

Cycle-type (group-theory)

Is there a formula for $\#$cycle types in $S_n$? Example: all cyle-types for $S_5$: \begin{align} &(5)\\ &(41)\\ &(32)\\ &(311)\\ &(221)\\ &(2111)\\ &(11111)\\ ...
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1answer
92 views

Suppose $G$ is a group and $N$ is a normal subgroup of $G$. Prove the following:

$(Nx)(Ny)=Nxy$ $N(x)(Nx^{-1})=(Nx^{-1})(Nx)=N$ $N(Nx)=(Nx)N=Nx$ Deduce that the right cosets of $N$ in $G$ form a group under the given multiplication.
1
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1answer
53 views

Maximal normal in solvable

I have problems proving this: "If G is a solvable group, then a maximal normal subgroup has prime index". I see that $\frac{G}{M}$ has to be simple. If I had that it has to be abelian too, I would ...
2
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3answers
111 views

If G is a group and A,B,C are subsets of G, prove that (AB)C=A(BC)

$A,B$ are defined as follows: $$AB = \big\{ ab \mid a\,\, \text{belongs to}\,\, A, \text{b belongs to B}\big\}\tag{1}$$ Where do I start?
1
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1answer
274 views

left regular representation of $G$ with respect to $g$

In the lecture, the prof used the fact that the ($\lambda_g$) left regular representation(??) is a bijection to prove cayley's theorem. Definition: left regular representation of $G$ with respect ...
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0answers
54 views

Is the group ring of a pro-finite group (semi)-hereditary?

It is well-known that a group ring of a finite group is semi-simple, and since profinite-groups are projective limits of finite groups, I am thinking per chance profinite groups still possess some ...
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97 views

Inner automorphism of a group-ring

Let $G$ be a group, $R$ a commutative ring with $1$ and $ u \in \text{N}_{\text{U}(RG)}(G) $ with the following properties: $ 1 \in \text{supp}(u) $ $ \text{supp}(u) \subseteq \Delta(G) $ $ ...
2
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1answer
54 views

Abelian group under the sum of spaces

Let $\cal W$ be the collection of subspaces of a vector space $V$. Question: Is $\cal W$ an abelian group under the sum of spaces?
2
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1answer
211 views

Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$

Suppose that $H$ and $K$ are normal subgroups of a group $G$ such that $HK=G$. I need to show that $G/(H\cap K)\cong (G/H)\times (G/K)$. So from the second isomorphism theorem we have that: ...
0
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1answer
132 views

Question about the number of endomorphisms/automorphisms of finite cyclic groups

Suppose $A,B,C$ are finite cyclic groups such that $A = B \times C$, where the orders of $B$ and $C$ are $p$ and $p^2$ respectively, where $p$ is a prime. What are the orders of $End(A)$ and $Aut(A)$? ...
6
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5answers
384 views

Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$

I tried using a contrapositive, ($m \neq n$ implies $\mathbb{Z}^m \ncong \mathbb{Z}^n$), and I think the problem is that there won't be a homomorphism, but did not get anywhere. Is there a better ...
1
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2answers
105 views

Prove that for every odd prime number $p$ there is a non-commutative group of order $p^3$ such that…

Prove that for every odd prime number $p$ there is a non-commutative group of order $p^3$ such that $a^p = e$, $\forall a \in G$.
6
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2answers
210 views

Given a group (finite or infinite), is there a way to find all its subgroups?

I know that for some small groups, one could just brute force all combinations of elements to find its subgroups, but is there some sort of an algorithm, at least for finite groups?
9
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3answers
1k views

What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be ...
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2answers
773 views

Is every group the automorphism group of a group?

Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$? EDIT: It has been pointed out that the answer ...
0
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2answers
2k views

Prove that any group $G$ of order $p^2$ is abelian, where $p$ is a prime number [duplicate]

Possible Duplicate: Showing non-cyclic group with $p^2$ elements is Abelian "Let $p$ be a prime number. Prove that any group $G$ of order $p^2$ is abelian. You may assume the fact that the ...
2
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3answers
884 views

What does “order” mean in group theory?

For example, if I have the question: "Find the primary decomposition of the abelian group $$ \mathrm{Aut}(C_{6125}). $$ Compute the number of elements of order 35 in this group." I know how to ...
0
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1answer
77 views

$K \subset S_4$ contains a 3-cycle and a 4-cycle $\implies K = S_4$

In my lecture notes of algebraic number theory they use the following claim to prove that the Galois group of $X^4 +X+1$ is the whole $S_4$ If $K$ is a subgroup of $S_4$ which contains a 3-cycle ...
2
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2answers
884 views

Defining the normalizer, showing its a subgroup and $|H| = |G:N_G(H)|$

"Let G be a finite group. For a subgroup $H \subset G$ defing the normalizer $N_G(H) \subset G$. Show that the normalizer is a subgroup, that $H \unlhd N_G(H)$ and that the number of subgroups $H'$ ...
0
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1answer
46 views

Is there a common notation for a group formed by appending a unit (eg. seconds) to some group?

Assume $(G,+)$ is an (abelian) group. And we have some unit $u$ (not the unit from ring theory, but some measurement unit like meters, seconds, etc.). Then with $G_u = \{k\ u \mid k \in G\}$ and ...
1
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1answer
115 views

semidirect product, split extension

I working on my thesis on semidirect products and splitting. I am trying to prove that if you assume that $G$ is a split extension of $N$ and $H$ then you can show that $G$ is a semidirect product of ...
4
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0answers
196 views

Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
53
votes
2answers
1k views

Reference request for tricky problem in elementary group theory

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of ...
4
votes
5answers
100 views

What's a proof that a set of disjoint cycles is a bijection?

Consider a function $f : D \to D$ (where $D$ is a finite set) so that for every $d \in D$, there is an integer $n$ so that $f(f(...(n\text{ times})...f(d)...) = d$. Prove that $f$ is a bijection. ...