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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A question on the order of an element involving relatively primes

This question is based on an exercise that comes from the second chapter of Malik's Fundamentals of abstract algebra which states as follows (I paraphrase): Let $(G, *)$ be a group and $x\in G$. ...
3
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2answers
745 views

Subgroups of Symmetric groups isomorphic to dihedral group

Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n>2$?
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1answer
57 views

How to show $G=G_m+G_n$ where $G$ is a finite abelian group?

Suppose $G$ a finite (additive) abelian group of order $|G|=mn$ and such that $gcd(m, n)=1$. Suppose $$G_m=\{g\in G: o(g)\mid m\}\quad \textrm{and}\quad G_n=\{g\in G: o(g)\mid n\}$$ How can I show $$G=...
6
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1answer
112 views

$G$ is torsion-free $[G:Z(G)]$ is finite $\implies$ $G$ is abelian ?

If $G$ is a a group having no non-identity element of finite order and $Z(G)$ , the center of the group , has finite index , then is it true that $G$ is abelian ?
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1answer
56 views

From Automorphism to abelian ness … in a finite group

Let $G$ be a finite group such that for any two non-identity elements $a,b$ in $G$ , there is an Automorphism of $G$ sending $a$ to $b$ , then is it true that $G$ is abelian ?
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2answers
72 views

$H$ of order $p$ normal in $G$ , g.c.d.$(|G|,p-1)=1$ , to prove that $H \subseteq Z(G)$

If $G$ is a finite group and $H$ is a normal subgroup of $G$ of order $p$(prime) such that g.c.d.$(|G|,p-1)=1$ , then how to prove that $H \subseteq Z(G)$ ? Please don't use any Sylow theorem or ...
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1answer
49 views

Constructing homomorphism from a special type of function

Let $f:G \to G$ be a function such that $f(a)f(b)f(c)=f(x)f(y)f(z) , \forall a,b,c,x,y,z \in G$ such that $abc=xyz=e$ ; then is it true that $\exists g\in G$ so that $h:G\to G$ defined as $h(x):=gf(x)$...
8
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1answer
103 views

Is the definition of stabilizer given at Planet Math really the currently accepted definition among group theorists?

According to Planet Math, given a group $G$ a set $X$ a subset $S \subseteq X$ and a group action $G \times X \rightarrow X,$ then the stabilizer of $S$ is define to be: $\{g \in G \mid gS \...
5
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0answers
273 views

What do group automorphisms fix? [closed]

I have often found it useful to sit and contemplate what kinds of elements, subsets, or structures do the automorphisms of an object fix or permute. Sometimes the observations do not have immediate ...
0
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1answer
74 views

Is an embedding of any group into itself always an automorphism?

I came across a question in chapter-8 The power of homomorphism (Visual group theory Book) which says that: Is an embedding of any group into itself always an automorphism? (Hint is that It is true ...
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1answer
192 views

Every subgroup of a cyclic group is characteristic (using lattice theory).

I want to show for $n\in\Bbb N$, which is not square-free, that every subgroup of $Z_n=\langle x\;|\;x^n\rangle$ is characteristic. But I want to show it in a convoluted way. Every automorphism $\...
4
votes
1answer
505 views

The alternating group is generated by three-cycles

Prove that, for $n \geq 3$, the three-cycles generate the alternation group $A_n$ Proof: We multiply on the left by 3-cycles to "reduce" an even permutation $p$ to the identity, using induction on ...
0
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1answer
31 views

Cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$

Let $m,n \in \Bbb Z^+$ such that m divides n. I'm trying to find the cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$. So, I think #$(m\Bbb Z_n)= \frac n m = k$. I tried to prove by it ...
2
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1answer
77 views

Classification of the decomposable primitive permutation groups

It is seen in comments here that the diagonal subgroup of the finite group $G \times G$ is core-free maximal iff $G$ is a nonabelian simple group. This gives examples of decomposable primitive ...
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3answers
146 views

homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ using $\phi(n)$=$2n$

A question from Visual group theory says : consider the homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ by $\phi(n)$=$2n$,where the group operation on $\mathbb Z$ is '+'. Would $\phi$ be ...
0
votes
1answer
50 views

Why $|G|$ even implies $|A(G)|$ also even?

Let $G$ be finite group with even order. Why has the set $A(G)=\{g\in G: g\neq g^{-1}\}$ an even number of elements?
2
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1answer
88 views

Why don't semi-direct products determine a group uniquely?

While reading some group theory notes I came up to this fact: Proposition: If $G$ is the inner semi direct product of $H,K$ ($G=HK$, $H\cap K=\left\{1\right\}$ and $H\unrhd G$) then $G\cong H\...
-1
votes
2answers
103 views

Prove that $a^n \cdot a^m = a^{n+m}$

Let $a$ be an element of a group $G$. Prove that $a^n \cdot a^m = a^{n+m}$ for any integers $m,n \in \Bbb Z$.
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2answers
95 views

Does there exists a homomorphism for any groups $G$ and $H$

This is a question from Exercise 8.2 of Visual Group Theory which says:determine whether true or false. For any group $H$ and $G$,there is some homomorphism from $H$ to $G$. For any groups $H$ and $...
3
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1answer
56 views

Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = \{...
4
votes
1answer
134 views

Map from real numbers to real numbers as abelian groups

What is an example of an abelian group homomorphism from $\mathbb{R}$ to $\mathbb{R}$, respecting addition, that is NOT a linear transformation? I'm trying to understand to what extent the scaling ...
3
votes
1answer
61 views

Is $\cos \frac{5\pi}{8}+i\sin\frac{5\pi}{8}$ in $U_8$, the multiplicative group of the $8$th roots of unity in $\mathbb{C}$?

I encountered this problem while reading Fraleigh, A First Course in Abstract Algebra, 4/e. (p.60 #19) Let $U_8$ be the multiplicative group of the $8$th roots of unity in $\mathbb{C}$. The question ...
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0answers
81 views

Is a primitive permutation group, indecomposable?

Is a primitive permutation group, indecomposable?
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1answer
289 views

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
12
votes
1answer
204 views

A group of order $120$ has a subgroup of index $3$ or $5$ (or both)

What I have tried that number of $2$-sylow subgroup can be $1,3,5$ or $15$.I have solved the problem when the number of $2$-sylow subgroup is $1,3,5$. But I am not able to solve it for $15$. Any help ...
3
votes
4answers
757 views

Prove that the symmetric group $S_n$, $n \geq 3$, has trivial center.

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let $\sigma*...
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1answer
92 views

What is the number of subgroups of order $7$?

$G$ be a simple group of order $168$. What is the number of subgroups of order $7$?
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1answer
130 views

Left Cosets of Cyclic Subgroup

Question from a GRE Math book that I'm having trouble understanding: Find the number of left cosets of the cyclic subgroup generated by (1, 1) of $$Z_{2} \times Z_{4}$$ where Zn denotes the cyclic ...
6
votes
2answers
162 views

Software or tool for investigating groups

I'm interested in software that has the ability to investigate finite groups. In particular, I'd like to be able to ask it questions like "What are the solutions to $x^3 = 1$?" (i.e. find cube roots ...
3
votes
1answer
78 views

Amalgams of two nontrivial group is trivial

I got this question in the book Tree by Serre. Let A=$Z$. $G_1=PSL(2,Q)$ and $G_2=Z/2Z$. We take $f_1: A\rightarrow G_1$ to be an injective (I do not know what is this injective map ?) and $f_2:\...
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2answers
76 views

Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let $...
0
votes
1answer
91 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, P_1,P_2,\...
2
votes
4answers
79 views

Calculate quotients of $\mathbb S_3$ and $\mathbb D_4$

Problem Calculate all the quotients by normal subgroups of $\mathbb S_3$ and $\mathbb D_4$,i.e., charactertize all the groups that can be obtained as quotients of the mentioned groups. For the case $...
0
votes
1answer
105 views

External semidirect product application

I am trying to find all normal subgroups of $\mathbb D_n$. I've read here Normal subgroups of dihedral groups that one could show the external semidirect product $(\mathbb Z/n\mathbb Z) \rtimes (\...
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1answer
69 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
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2answers
65 views

$G=\{f_n(x):n\in \mathbb{Z}\}$ is cyclic

Define $f_n(x)=x+n \;\;\forall n \in \mathbb{Z}$. Let $G=\{f_n:n\in \mathbb{Z}\}$. I proved that $G$ is a subgroup of $S_\mathbb{R}$ ($f_n$ is a permutation of $\mathbb{R}$), and now I am trying to ...
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2answers
93 views

Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
2
votes
1answer
103 views

left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
0
votes
2answers
147 views

Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
2
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1answer
67 views

What can you say about the order of this element?

I am self-studying algebra and encountered the following problem: If $b^{-1}ab=a^2$ and $c^{-1}ac=a^3$ and $b,c$ has orders $4$ and $3$ respectively, what can you say about the order of $a$? In ...
2
votes
1answer
51 views

homomorphism between diedral group $D_3$ triangle isometries and $S_3$ identification problem

My question deals with the dihedral group $D_3$ of equilateral triangle 123 (1 top vertex, 2 bottom right vertex, 3 bottom left vertex). R1 is the counterclockwise rotation of 120 degrees. R2 is the ...
12
votes
2answers
443 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
1
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1answer
98 views

Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] (...
9
votes
1answer
159 views

Cardinality of $\text{Aut}(G\times G) $

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
2
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1answer
558 views

Solvable implies quotient group is solvable: Proof check.

I'd like to check the veracity of my proof. I've seen several proofs using different methods (some I'm allowed to use with lots of element-pushing and others using ideas I'm not allowed), but none ...
4
votes
2answers
780 views

Checking understanding on proving uniqueness of identity and inverse elements of a group.

Sorry for such a trivial question, but just wanted to check my understanding. When proving a statement, for example, that the inverse of a group element is unique (in elementary group theory) one ...
2
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0answers
100 views

Conditions for a group to be isomorphic to semidirect product of its subgroups

Let $G$ a group and $N$ a normal subgroup of $G$. If $G$ it have a subgroup $H$ s.t. $H \cap N $ is the trivial subgroup and $H$ is isomorphic to $G/ N$ then $G$ is isomorphic to $N\rtimes H$. Could ...
2
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0answers
42 views

No proper subgroup of finite index [duplicate]

Show that $(\mathbb{Q},+)$, the group of rational numbers under addition, has no proper subgroup of finite index. Can someone please provide a proof!
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0answers
39 views

Proving $|G|=pq$ and $p>q$ , $q$ does not divide $p-1$ $\implies$ $G$ is cyclic , without using Cauchy's and Sylow's theorems [duplicate]

Without using Cauchy's or Sylow's theorems , can we give a proof of the result that "If $ p,q$ are primes such that $p>q$ and $q$ does not divide $p-1$ , then any group of order $pq$ is cyclic " ?
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0answers
47 views

divisible subgroup without axiom of choice

the theorem asserting that the divisible subgroup of an Abelian group is a direct summand depends on Zorn's lemma. in ZF without AC is there a construction which yields a model of an Abelian group ...