0
votes
3answers
44 views

If $H$ and $K$ are subgroups of G then $H \times K$ is a subgroup of $G \times G$

I know that if $H$ and $K$ are subgroups of $G$ then $HK= \{ hk \mid h \in H , k \in K\}$ is not necessarily a subgroup of $G$, this requires that $HK = KH$. But it follows that if $H$ and $K$ are ...
-2
votes
0answers
41 views

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
1
vote
1answer
40 views

Irreducible action of a group on a set

Let $p$ be a prime. I am solving a problem and I am told that I should use that the action of $\text{SL}_2(p)$ on $\mathbb{F}_p^2$ is irreducible. But I don't know what this means?
2
votes
1answer
30 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
3
votes
2answers
102 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
1
vote
0answers
33 views

Can the derived subgroup be realized as an intersection of stabilizers?

For any group $G$, we have that $Z(G)=\bigcap_{x\in G}C_G(x)$ and that each $C_G(x)$ is the stabilizer of $x$ when $G$ acts on itself by conjugation. Is there a similar representation for $G'$? That ...
1
vote
1answer
43 views

Finitely generated, periodic group such that each conjugacy class is finite must be finite?

This is essentially a repeat of this question. However, the OP didn't seem to put in any work towards a solution and didn't provide context. Nevertheless, I'm still interested in the solution, and I ...
-2
votes
0answers
26 views

Number of normal subgroups of a non-abelian group of order 21 [on hold]

How many normal subgroups can a non-abelian group G of order 21 have other than the identity subgroup {e} and G? a)0 b)1 c)3 d)7
2
votes
1answer
28 views

Constructing an indicator function from a braid group which represents 'all strings have returned to their initial position'.

TL;DR Is there a well-defined closed formula from the braid group $B_n$ to $\{-1,1\} \left(\text{ or }\{0,1\}\right)$ which represents whether all the strings have returned to their initial ...
0
votes
1answer
28 views

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. [on hold]

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. I know the elements of Z/10Z are {1,3,7,9}, is that the same for ...
5
votes
3answers
104 views

Finitely-generated group such that all (non-trivial) normal subgroups have finite index implies all (non-trivial) subgroups have finite index?

Let $G$ be a finitely generated group such that every non-trivial normal subgroup has finite index. Does it follow that every non-trivial subgroup of $G$ has finite index? This question arose as ...
0
votes
2answers
66 views

We have a map $g : \mathbb{Z}/24\mathbb{Z} → \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} $. What is the kernel of $g$?

We have a map $g : \mathbb{Z}/24\mathbb{Z} → \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$ given by $g(x+24\mathbb{Z}) = (x + 6\mathbb{Z}, x + 4\mathbb{Z})$. What is the kernel of $g $? In ...
2
votes
0answers
61 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
2
votes
1answer
55 views

Write cyclic groups of order $p^n$ in terms of simple groups

Some say that studying simple groups helps you understand the structure of non-simple groups. How can I write in terms of simple groups $\mathbb{Z}_{p^n}$? Eg. $\mathbb{Z}_9$
0
votes
0answers
60 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
0
votes
1answer
37 views

Subgroups of $\mathbf{Z}/20\mathbf{Z}$ using Lagrange's Theorem

According to Lagrange's Theorem, what are the possible sizes of the subgroups of $\mathbf{Z}/20\mathbf{Z}$? I have no idea how to go about answering this. I have a feeling that I should be ...
10
votes
1answer
90 views

Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
-2
votes
0answers
42 views

Basic group theory [closed]

Suppose that $H$ is a subgroup of $G$ such that whenever $Ha$ is not equal to $Hb$ ... then $aH$ is not equal to $bH$ . Prove that $gHg^{-1}$ is a subset of $H$ for all $g$ belonging to $G$.
1
vote
1answer
48 views

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
2
votes
2answers
45 views

How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
1
vote
1answer
47 views

I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
0
votes
1answer
53 views

Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
0
votes
1answer
45 views

Group theory problem

I am asked to prove "Show that if $${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 ...
1
vote
0answers
24 views

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
1
vote
0answers
38 views

Show that an algebra is a linear algebraic subgroup in $GL(A)$, $A$ being a finite dimensional algebra over $\mathbb C$

Let $A$ be finite dimensional algebra over $\mathbb C$ with unit 1. Let $G$ be the set of all $g \in A$ such that $g$ is invertible in $A$. For $z \in A$ let $L_a \in$ End$(A)$ be the operator of left ...
2
votes
1answer
32 views

For a group $G$ acting transitively, is $G_{\alpha}$ contained in stabilizer of some block $\Delta$ if $\alpha \in \Delta$?

This question refers to permutation groups, in particular the primitive ones, and block systems. Let $G$ be a finite group acting on a set $\Omega$, and consider some partition $\Delta_1 \cup \ldots ...
4
votes
2answers
105 views

A normal subgroup problem

Let $G$ be a group in which, for some integer $n>1$, $(ab)^{n}=a^{n}b^{n}$ for all $a,b \in G$. Show that $G^{(n)}=\{x^{n} \mid x \in G\}$ is a normal subgroup of $G$. $G$ could be easily ...
3
votes
3answers
148 views

Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
1
vote
1answer
42 views

Show for each $c$, $\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$ is an abelian group under multiplication of congruence classes

Show for each $c$, the set $$\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$$ is an abelian group under multiplication of congruence classes.
0
votes
1answer
45 views

generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
4
votes
1answer
57 views
+100

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
2
votes
1answer
30 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
3
votes
3answers
222 views

What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...
9
votes
4answers
153 views

Does every group act faithfully on some group?

Cayley's Theorem shows that every group acts faithfully on some set. In other words, one can find an injective group homomorphism $\sigma: G\to S_{A}$ where $S$ is the set of all bijections on some ...
4
votes
2answers
118 views

If a group has no maximal subgroups then all elements are non-generators? Frattini subgroup characterization

This question is the last leg of an exercise I've been working on in which we characterize the intersection of all maximal subgroups as the subgroup of all non-generators. I've already shown that if a ...
3
votes
1answer
46 views

calculating signature and showing group homomorphism

I got this question in my group theory book. I think I understand the theory behind it but can't seem to use it to get a solution im happy with. Let $V = M_2(\mathbb F)$. For $x,y \in V$ define ...
3
votes
1answer
50 views

Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
7
votes
1answer
133 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
0
votes
0answers
45 views

Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
0
votes
0answers
32 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
2
votes
1answer
55 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
0
votes
2answers
34 views

order of dihedral

I am learning abstract algebra, and I don't quite understand the order of the symmetry of dihedral. When you look at a squares, I agree that there will be 8 symmetry. But all the operations have cycle ...
0
votes
1answer
46 views

The relation between orders in a group

G is a group and N is a normal subgroup of G.what is the relation between the order of $x$ and $x.N$?
2
votes
3answers
99 views

Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9: Let $G$ be the set of all $2 \times 2$ matrices $ \left( {\begin{array}{cc} a & b \\ c & d \\ ...
3
votes
2answers
34 views

Prove that the symmetric group on $n$ letters, $S_n$, has order $n!$.

Here's my proof in which I've used another theorem to prove this one. I want you suggest me another proof without using this theorem, please. Proof: By the theorem Cardinality of set of injections, ...
4
votes
1answer
116 views

Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
1
vote
1answer
51 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
1
vote
1answer
47 views

number of cycles in a permutation

I have given a permutation let 2 3 1 5 4 that is if initially my string is 1 2 3 4 5 the after one permutation it will become 1 2 3 4 5 3 1 2 5 4 that is the number in first position will go ...
4
votes
2answers
50 views

How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
-1
votes
0answers
35 views

Behavior of groups under extension

We know that an extension of a solvable group by a solvable group is solvable. Similarly we can find other properties of group extensions here Can someone provide a reference to these statements where ...