Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [group-theory]

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 is the study of groups.

0
votes
1answer
17 views

Monic polynomial with coefficients in $\mathbb F_2$ whose companion matrix is invertible and has largest possible multiplicative order

Let $a_0,...,a_{n-1}\in \mathbb F_2$ be such that the companion matrix (https://en.wikipedia.org/wiki/Companion_matrix) of the monic polynomial $a_0+a_1X+...+a_{n-1}X^{n-1}+X^n\in \mathbb F_2[X]$ is ...
0
votes
0answers
8 views

Composition factors above a subnormal subgroup

Suppose $G$ has a composition series and let $H$ be a subnormal subgroup of $G$. Then $H$ is a term of a composition series of $G$. I know that the number of terms above $H$ in any composition series ...
0
votes
1answer
22 views

Deduce that the symmetry group of the dodecahedron is a subgroup of $S_5$ of order 60.

Let D be a regular dodecahedron. It is possible to inscribe a cube on the vertices of D thus: (a) Prove that one can inscribe exactly 5 such cubes inside D. (b) Deduce that any rigid motion of D (i....
3
votes
0answers
44 views

Exponent of the group $GL(n,\mathbb F_2)$

Let $A\in GL_n(\mathbb F_2)$ be an element of order greater than or equal to $2^n-1$ . Then is it true that order of $A$ is $2^n-1$ ? I know that $|GL_n(\mathbb F_2)|=(2^n-1)(2^n-2^2)...(2^n-2^{n-1})$...
6
votes
2answers
36 views

Elements of $S_n$ can be written as a product of $k$-cycles.

Let $k\leq n$ be even. Prove that every element in $S_n$ can be written as a product of $k$-cycles. I really have no idea how to go about this. My initial intuition was to proceed by induction first ...
0
votes
0answers
19 views

Partition of a conjugacy class to conjugacy classes of a normal subgroup

Let $G$ be a group, $H$ be a normal subgroup of $G$, and $O$ a conjugacy class of $G$ contained in $H$. Consider $O = \cup_{i = 1}^{n}O_i$ the partition of O into conjugacy classes of $H$. Show that ...
0
votes
1answer
23 views

Some results given that $|G| = nm$

Let $G$ be a finite group such that $|G| = nm$. Suppose $x\in G$ has order $n$ and let $\sigma_x\in S_G$ be the permutation such that $\sigma_x(g)=xg$ for every $g\in G$. Note that $\sigma_x$ is a ...
1
vote
1answer
18 views

Understanding conjugacy classes in $SL_{2}(\mathbb{F}_{q})$

I am trying to calculate the conjugacy classes of the group $SL_{2}(\mathbb{F}_{q})$, with the help of the knowledge of conjugacy classes of $GL_{2}(\mathbb{F}_{q})$. I am using two of the following ...
-1
votes
1answer
15 views
-1
votes
1answer
21 views

If $A\cong B$, $C\cong D$ with $D\trianglelefteq B$, $C \trianglelefteq A$ then $A/C\cong B/D$.

Suppose that $A,B,C,D$ are groups such that $A\cong B$, $C\cong D$ with $D\trianglelefteq B$, $C \trianglelefteq A$. Prove that $A/C\cong B/D$. Proof: Suppose that $f:A\to B$ and $g:C\to D$ are ...
3
votes
1answer
45 views

Abelian group operation on $(0, 1)$

My Question is: is there a binary operation on $(0, 1)$ that makes this set into an abelian group in which the inverse for any $x$ is $1-x$? My approach to this was applying $f(x)=\pi(x-0.5)$ (which ...
0
votes
1answer
36 views

Prove that if $|a| = m$ and $|b| = n$, then $\exists g \in G$ s.t $|g| = mn$

In the book of Algebra by Hungerford, at page 36, question 2, it is asked that For the case where $(m,n) = 1$, I have done the following: Observe that $$(ab)^{mn} = a^{mn} b^{mn} = e.$$ Now, let $...
2
votes
2answers
32 views

Definition of graded abelian group

I am reading Homology Theory by Vick, and in this book a graded abelian group $G$ is defined to be a “collection of abelian groups {$G_i$} indexed by the integers with component-wise operation”. What ...
0
votes
0answers
37 views

Prove that $\left(\forall\ a \in A \right)\ aTe = eTa = e$

Let $\left(A,\ *, T\right)$ be a Ring and $e$ be its identity element. Is it possible to prove the following statement without using the regularity of $aTe$ in the underlying group $\left(A,\ *\right)...
0
votes
0answers
19 views

How to determine basis functions partners of $x$, $x^2$, and $yz$ in $D_6$?

I have a character table for $D_6$ and I am trying to understand how to find the partners of basis functions. I am starting with $x$, $x^2$ and $yz$. I am currently working on the $x$ function but am ...
1
vote
1answer
29 views

Show that $\operatorname{GL}_2 (\mathbb{F}_3)/\{\pm I_2\} \cong S_4$

Let $\mathbb{F}_3$ be a field with three elements and let $V = \mathbb{F}_3^2$. Let $\alpha,\beta,\gamma$ and $\delta$ be the four one-dimensional subspaces spanned by $\begin{bmatrix}1\\0 \end{...
0
votes
0answers
13 views

Index of intersection of subgroups [duplicate]

Let $G$ be a group with subgroups $H$ and $K$ with index $r=[G:H]$ and $s=[G:K]$. Show that: $[G:H\cap K]\leq rs$ I know it works with Lagrange, but I don't know exactly how.
0
votes
1answer
18 views

In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ ...
0
votes
1answer
26 views

What is the definition of the least common multiple of elements of a group rather than their orders?

In an answer to my question here Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?, it is said that Let $G$ be a group, $a\in G$...
1
vote
0answers
23 views

Covering of a topological group

Let $X$ be a topological group with a locally path-connected, path-connected covering $(\tilde{X},p)$. If we fix an $u\in p^{-1}(e)$, we should be able to deduce a unique group structure for $H=\tilde{...
1
vote
1answer
29 views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In this question Prove that the product $NK$ of two normal subgroups $N$ and $K$ of a group $G$ is a normal subgroup of $G$, and $NK=KN$., it is proved that $NK=KN$ if both $N$ and $K$ are normal ...
1
vote
1answer
22 views

Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let x be an element of order r of a group G, and let y be an element of G' of order s. What is the order of $(x, y)$ ...
0
votes
0answers
19 views

What is meant by “the group $\langle a \rangle$ acts on the set $A$ by conjugation?”

A proof for the fact that an abelian group of order 15 must be cyclic states is given here: https://math.stackexchange.com/a/208801/115703 In this proof, what is meant by "The group $\langle a\...
0
votes
0answers
13 views

How to compute $\operatorname{Ker}(\phi )$ and $(\phi )(25)$ for $(\phi ):Z\rightarrow Z_{7}$ such that $\phi(1)=4$? [duplicate]

I understand I must find some element that map to identity but what condition $\phi(1)=4$ is related ? and how to compute $(\phi )(25)$? I must find mapping from 25 to $Z_{7}$ isn't it ? is that 4 ?
4
votes
1answer
34 views

why composition series for groups are of finite length?

def : a composition series of a group G is a subnormal series of finite length $$ 1 \vartriangleleft H_1 \vartriangleleft H_2 \vartriangleleft \cdot \cdot\cdot\cdot\cdot\cdot\cdot \vartriangleleft ...
2
votes
1answer
38 views

Prove $[G:H]=[G':H']$ where $H$ and $H'$ are corresponding subgroups.

For a surjective homomorphism $\varphi: G \to G'$ where $\ker \varphi \subseteq H$, $H$ is a subgroup of $G$ that corresponds to $H'$, a subgroup of $G'$, the correspondence theorem implies $$\varphi(...
1
vote
1answer
27 views

Torsion elements of a group aren't necessarily a subgroup [duplicate]

In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup. I knew that the set ...
0
votes
0answers
21 views

Structure constants and matrix representations

I have a doubt about the relation between the structure constants of a Lie group and the representation of its algebra. Let's see an example: For $SU(3)$ the number of generators is 8 and we write ...
1
vote
0answers
24 views

A proof for the statement: “Let $G$ be a group, with $H$ normal within it such that $|G| = r|H|$. Show that $g^r \in H$ for all $g \in G$.”

Here is my proof: If $|G| = r|H|$, then $[G:H] = r$, which means that $G/H$ has $r$ elements in it, each one corresponding to one of the $r$ cosets of $H$. Pick a system of representatives $e, x_1, ...
0
votes
0answers
22 views

A subgroup of abelian torsion group with infinite exponent

Take an abelian torsion group, with infinite exponent, say, G=C2+C4+C8+..+C2^k+.., where Cn is cyclic of order n. What is the subgroup S that is the intersection of all subgroups of G with infinite ...
0
votes
0answers
32 views

Question from Algebra: Chapter 0 by Aluffi [duplicate]

Let K be a normal subgroup in finite group G. Assume |K| and [G:K] are relatively prime. Is K characteristic in G? I am self-studying through this book. Please only provide very small hints.
0
votes
0answers
16 views

Arguing that $(A\cap B)/N = A/N \cap B/N $

Let $N\leq A, B\leq G$ where $N\trianglelefteq G$ so that $G/N$ is a group. With correspondence theorem, I am trying to show that the join and intersection of $A$ and $B$ has a unique correspondence ...
0
votes
1answer
27 views

Problem about index of proper nontrivial subgroup

Show that a finite simple group $G$ of order $\geq d!$ can not have a proper nontrivial subgroup of index $d$. Remark: I guess that the condition of this problem is a bit incorrect, namely we need to ...
6
votes
1answer
44 views

Validation for a conjecture about Chinese Remainder Theorem for groups

I was wondering if the following statement is true: Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $i\neq j$. Then $G/H_1\cap H_2...\cap H_n\cong G/H_1 \times......
1
vote
1answer
31 views

Subgroups of symmetry Group

I'm reading Matrix Groups for Undergraduates by K.Tapp. In chapter $3$ he defines symmetry group as the group of all isometries of $\mathbb{R^n}$ that carry $X\subset\mathbb{R^n}$onto itself: $Symm(...
-1
votes
0answers
34 views

Help proving Sylow's Theorem order 58

Prove that every group of order 58 is not simple. So I know that 58 = 2 ⋅ 29. I assume G is simple. I'm having trouble using the Sylow Theorems to show that this is not Simple. In particular, ...
0
votes
1answer
20 views

Simple nonabelian where orbits have length at least 3

If $G$ is a simple nonabelian group and $e\neq x \in G$, show that $x$ must have at least 3 conjugates (including itself). My attempt: Suppose $G$ acts on itself by conjugation, i.e. for any $g\in G$,...
0
votes
1answer
16 views

Proving that if $G$ is a group of order $p^2$ where $p$ is prime, then $G$ is either $\mathbb{Z}_p \times \mathbb{Z}_p$ or $\mathbb{Z}_p$

Prove that if $G$ is a group of order $p^2$ where $p$ is prime, then $G$ is either $\mathbb{Z}_p \times \mathbb{Z}_p$ or $\mathbb{Z}_{p^2}$ I'm aware that this question has an answer here : There ...
0
votes
2answers
48 views

Handling the real numbers, multiplication, and zero as a group

A group can be formed using a set and a binary operator on elements of a set. Consider {$\mathbb{R}$, x}, the real numbers $\mathbb{R}$ and the multiplication operator x. There are nine product rules. ...
1
vote
1answer
29 views

Ring structure on power set of a set.

Let $A$ be a non empty set. Let $P(A)$ denote the power set of $A$. $P(A)$ can be given a group structure in multiple ways. 1. Using Disjoint union a group operation Source may be worth noting in ...
3
votes
1answer
33 views

Proof Check: Let $f_{0}(x)=\frac{1}{1-x},$ and $f_{n}(x)=f_{0}(f_{n-1}(x))$ for any positive integer $n$. Find $f_{2018}(2018)$

Let us denote by $F$ the set of real-valued functions of a real variable such that $$F=\left\{\frac{1}{1-x},\frac{x-1}{x},x\right\}.$$ We shall prove that this set forms an abelian group under the ...
6
votes
1answer
33 views

Sum of values of an irreducible character is non-negative integer

I am trying to prove the following fact: If $G$ is a finite group, $\chi$ is a complex irreducible character of $G$ and $\{g_1,\cdots,g_r\}$ is a complete set of representatives of conjugacy ...
5
votes
1answer
33 views

Let $G$ be a finite matrix group in $GL_2(Q)$ such that every matrix $A\in G$ has integer entries. Prove $A^{12}= I$ for each $A$.

Let $G$ be a finite matrix group in $GL_2(Q)$ (general linear group of $2$ by $2$ matrices with rational entries) such that every matrix $A\in G$ has integer entries. Prove that $A^{12} = I$ for every ...
3
votes
1answer
40 views

If $S$ is subnormal in $G$, $S$ is simple and nonabelian and $S \subseteq H \subseteq G$, then $S \subseteq \textrm{Soc}(H)$.

I suspect I have missed some easier way to show this claim, and there might be a mistake in my approach. I know this is a very lengthy proof, but I spent a lot of time on trying to solve this problem ...
1
vote
0answers
41 views

Structure constants as functions

I wanted to know if the structure constants of a Lie group can be function, or if the have to be proper numbers. I think that because of Lie algebra is obtained by derivation of group's elements and ...
1
vote
1answer
41 views

Lie group O(3, 2)

I've looking for the matrices of the O(3, 2) Lie group or, instead of them, the structure constants of its algebra $\mathfrak o(3, 2)$ but I've not been successfull. Does anyone know them or can ...
0
votes
0answers
26 views

Index of congruence subgroups in $GL_2(\mathbb{Z}_p)$ modulo their centers.

Let $\Gamma_i$ be the set of matrices in $GL_2(\mathbb{Z}_p)$ which are congruent to $1$ modulo $p^i$, that is they are the congruence subgroups. I know that $\Gamma_i$ is a pro-$p$ group and $\...
1
vote
1answer
20 views

Fixing the representative of a coset for a function.

I am working on through a proof from lecture. Given that $K\leq H\leq G$, we are constructing a bijection $$f: G/H\times H/K\rightarrow G/K$$ Since $G/H$ is the set of cosets, we can fix the ...
1
vote
0answers
26 views

Suggested Reading? Applications of the Krull-Schmidt Theorem for Groups

I'm currently reading about the Krull-Schmidt Theorem for groups, and am wondering how I can learn more about its role in group theory. Thus far I have read the proof sketch given in Hungerford's ...
0
votes
1answer
24 views

Can an automorphism of a group which stabilizes a subgroup fail to restrict to an automorphism of the subgroup?

Does there exist a group $G$, a subgroup $N\le G$, and an automorphism $\alpha$ of $G$ such that $\alpha(N)\subset N$, but $\alpha$ does not restrict to an automorphism of $N$? Equivalently, can ...