1
vote
0answers
47 views

How can I construct a surjection with a kernel $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$?

How can I construct a surjection from a group $G$ to $S_3$ with the kernel $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ ? I know how to construct the surjection from $S_4$ to $S_3$ with ...
0
votes
0answers
24 views

Find the group of automorphisms of the Petersen graph [duplicate]

I came across a problem which said "find the group of automotphisims of the Petersen Graph." How should I go about doing this?
1
vote
2answers
37 views

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 $x\in G$. ...
1
vote
1answer
28 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 $n>2$?
0
votes
1answer
41 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 ...
2
votes
0answers
62 views

What do group automorphisms fix?

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
votes
1answer
27 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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
79 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 ...
0
votes
1answer
26 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 ...
1
vote
3answers
93 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
43 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?
1
vote
2answers
80 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 $m,n \in \Bbb Z$.
1
vote
2answers
49 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
votes
1answer
45 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 ...
9
votes
2answers
101 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 ...
-2
votes
1answer
71 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 $7$?
1
vote
1answer
35 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 ...
0
votes
1answer
31 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 ...
1
vote
2answers
62 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
33 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
48 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
votes
1answer
63 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 ...
6
votes
1answer
277 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
vote
1answer
42 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
121 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 $|\text{Aut}(G\times G)|$ ?
1
vote
1answer
27 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 ...
2
votes
0answers
50 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 ...
3
votes
1answer
50 views

Something that is true for every element of $\text{Sym}(\Bbb{N})$

I'm trying to prove that: Every element of $\text{Sym}(\Bbb{N})$ can be written as $f\circ g$ for some $f,g\in \text{Sym}(\Bbb{N})$ with $f^2=g^2$. But I can't even prove this for ...
3
votes
1answer
136 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
-4
votes
1answer
100 views

For an associative binary operation with identity, the set of invertible elements forms a group

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
2
votes
2answers
30 views

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$ Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} ...
3
votes
1answer
45 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
1
vote
2answers
39 views

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$. Attempt: $|G|=168=2^3.3.7$ Then number of sylow $7$ subgroups in $G ...
0
votes
1answer
27 views

Proof the existence of a normal subgroup

Let $K$ be a normal subgroup of $H/N$, and $N$ be a normal subgroup of $H$. Show that there is $M \lhd H$ such that $N \subset M$ and $K=M/N$. I have some difficulties to prove it.
2
votes
3answers
88 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
1
vote
2answers
61 views

The multiplicative group of all the $2^n$-th roots of unity

Consider the multiplicative group $G$ of all the (complex) $2^n$-th roots of unity where $n=0,1,2...$. Which of the following statements are true? Every proper subgroup of $G$ is finite, $G$ has a ...
2
votes
2answers
45 views

Examples for infinite Hamiltonian group

During teaching some basic concepts about a Hamiltonian group, I was asked about an infinite sample. According to what D.J. Robinson cited, we have a very good ...
3
votes
0answers
33 views

If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
4
votes
1answer
60 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
6
votes
2answers
142 views

Self studying higher mathematics?

I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they ...
2
votes
1answer
41 views

Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
2
votes
1answer
49 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
2
votes
2answers
41 views

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean?

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean? I apologize if this is too basic, but I haven't come across such an expression anywhere in my book. Also, ...
3
votes
2answers
36 views

Question about the symmetric group

How do I prove that if $f\in S_k$ and $f^n=f^m=Id$, then $f^d=Id$ where $d=gcd(n,m)$? I tried writing $f^{dn_1}=f^{dm_1}=f^0$ but this does not lead anywhere. I think I should use that $n_1$ and ...
1
vote
0answers
29 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
6
votes
4answers
151 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
1
vote
2answers
60 views

Semidirect Product variations

Is there two semidirect products of order $16$ with $C_2$ normal subgroup? How many groups of the form $C_4 \times C_2$ : $C_2$ are there ? Is it one expression for two groups ? or more?
1
vote
1answer
111 views

Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
2
votes
1answer
43 views

The only group of order $255$ is $\mathbb Z_{255}$ ( Using Sylow and the $N/C$ Theorem)

Let $G$ be a group such that $G =255 = 3.5.17$. Let $H$ be a sylow $17$ sub group of $G$. Then, by Sylow's theorem : the number of sylow $17$ sub groups can be $n=1,18,35,...$ . Since, $n$ should ...