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.

learn more… | top users | synonyms (2)

2
votes
2answers
291 views

If I have the presentation of a group, how can I find the commutator subgroup of it?

I have the group given by the presentation $G= \langle a,b\mid a^2,b^2\rangle$ How can I in general find $G',G/G',G''$ ? thanks for any hints.
2
votes
1answer
108 views

Are subgroups kernels of some $1$-cocycles?

If an $1$-cocycle $\sigma :G\to K$ has image in a group on which $G$ acts, then the set of element of $G$ that is mapped to $\text{$e_K$}$ by $\sigma$ forms a subgroup of $G$. This could be proved by ...
3
votes
1answer
74 views

If $gxg^{-1}\in X$, $\forall x\in X$, then $\langle X\rangle$=$\langle X\rangle^G\lhd G$

Well in this exercise i don't get more information but what is in the title. I don't understand how this set would even look like $\langle X\rangle$. I'm gessing $\langle X\rangle=\{ x^n| n\in \mathbb ...
1
vote
1answer
83 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
3
votes
1answer
130 views

Questions about the Space of Matrix Coefficients

Apologies in advance for the basic question: In reading up on representation theory, I came across a confusing definition for the $M(\rho)$, the space of matrix coefficients of a representation $(G, ...
3
votes
2answers
128 views

Counter examples on Categories

I'm reading Categories for the Working Mathematician by Saunders Mac Lane. At the section 5 from chapter 1, for a fixed category, he claims that every arrow with right inverse, is epic (right ...
1
vote
2answers
48 views

A translation and a negation in $\mathbb{C}$ generate the infinite dihedral group.

I'd like to show that the linear functions $$ \varphi(z) = z+b, \;\;\; 0\neq b\in \mathbb{C}$$ $$ \psi(z) = -z+c, \;\;\; c\in \mathbb{C}$$ generate, under composition, a group isomorphic to ...
0
votes
1answer
45 views

Suppose G is a finite and simple group which acts transitively on S. Given that $k \equiv |S| > 1$, prove $|G|$ divides $ k!$

I think they key point here is to prove that $G$ must be isomorphic to a subgroup of $S_k$ then we would be done. I am quite lost in trying to do so. Since $G$ acts transitively on $S$, can't we just ...
2
votes
1answer
57 views

Is $G $ contained in $ G *_H K $ if $ H\rightarrow G $ and $ H\rightarrow K$ are injections?

Given injections $ H\rightarrow G $ and $ H\rightarrow K $, is the canonical morphism $ G\rightarrow G *_H K $ of G into the free product with amalgamation also injective?
2
votes
1answer
84 views

Schmidt group and sylow subgroup

Let $G$ be Schmidt group. If $Q=\langle a\rangle$ is $q$-subgroup of $G$, then $a^q \in Z(G)$
0
votes
2answers
91 views

how the set of all inner automorphisms of each group $G$?

1] show the set of all inner automorphisms of each group $G$ is a normal subgroup of the group of all automorphisms $G$?
3
votes
1answer
64 views

Is meta-$p$-nilpotent for all $p$ the same as meta-nilpotent?

I've been working on understanding solvable groups one prime at a time, and it has been very helpful. Perhaps this question is enjoyable to more than just me: Definition: A finite group is said to be ...
2
votes
1answer
339 views

Number of edge colorings in a tetrahedron with three colors. Is my solution correct?

I've tried to count rotationally distinct edge colorings (both proper and improper) in a regular tetrahedron with three colors. Could you take a look if it's correct? First the improper colorings. ...
1
vote
2answers
105 views

Metanilpotent groups and saturated formation

Class of all metanilpotent groups is a saturated formation ? How do I prove
3
votes
2answers
105 views

What is a group action in simple English? [duplicate]

What does it mean for a group to act on a manifold? Or what does it mean for a group to act on a vector space? Are rotations of an objection considered group actions? Can other things "act" on various ...
2
votes
1answer
103 views

Is this graph coloring problem solved correctly?

On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done. They ...
2
votes
1answer
75 views

Group representation in MAGMA

I worked with MAGMA online and studied the handbook, but I still do not know how to print the Cayley table. It is possible to print the elements, but the representation is somewhat unuseful. It ...
0
votes
1answer
86 views

Addition on elliptic curves

assume $a$, $b$ are two integer numbers, and $G$ is a basepoint in an elliptic curve. Is $(a+b)G$ equal to $aG+bG$ or not?
1
vote
1answer
95 views

Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism

Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism given that there is bijective and holomorphic function $f$ from region $D$ onto region $D'$ (in other words, $f^{-1}$ is also holomorphic) any idea ...
3
votes
2answers
329 views

Index of maximal subgroups of $p$-solvable groups

If $G$ is finite $p$-solvable group (every chief factor has order either a power $p$ or relatively prime to $p$) must every maximal subgroup have index either a power of $p$ or relatively prime to ...
16
votes
1answer
498 views

What is computational group theory?

What is computational group theory? What is the difference between computational group theory and group theory? Is it an active area of the mathematical research currently? What are some of the ...
6
votes
3answers
420 views

Is there a Dihedral group of order 4?

if i use the notation $D_{2n}$ does $D_4$ makes sense ? if i showed that a group $G$ is isomorphic to $H \times D_4$ where H is a group , then $G$ is not a group ? why am i ask this ? because ...
2
votes
3answers
1k views

How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group" ...
2
votes
1answer
118 views

Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.

Prove $D_{8n} \not\cong D_{4n} \times Z_2$. My trial: I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows: Suppose $D_{4n}$ is isomorphic ...
3
votes
1answer
211 views

Galois group of $K(X)/K$

Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$. Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq ...
1
vote
1answer
52 views

Automorphism that saves all subgroups of gruop.

Let $h\in$Aut($G$) so that it saves subgroups: $h(U)=U$ for each subgroup $U$ of $G$, and $\alpha$ is any automorphism. Is it true that $\alpha h \alpha^{-1}$ also saves subgroups?
0
votes
2answers
81 views

finite group and $M$ is a maximal subgroup

Let $G$ be a finite group and $M$ is a maximal subgroup of $G$. Prove $\forall g \in G$, $M^{g}$ is a maximal subgroup of $G$.
0
votes
0answers
51 views

Examples of group morphisms which are not linear applications? (In $\mathbb R$.) [duplicate]

I was wondering whether there are group morphisms in $\mathbb R$ which are not linear applications. I would have guessed that it exists but I cannot think of an example. Would someone have some ...
10
votes
3answers
282 views

Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$

If the order of $G$ is $p^2$ then how do I show that $G$ is isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p\times\mathbb Z_p$.
1
vote
0answers
64 views

Chernikov and Sylow

Let $G$ periodic (infinity). Let $\mathbb{P}$ is a set primes numbers, $\pi \subseteq P$ and $\pi ^{\prime }=P-\pi$. Let $O_{\pi }\left( G\right) =\left\langle N \ ;\ N\unlhd G, N \ \ is \ \pi ...
0
votes
2answers
222 views

Order of the group generated by two matrices

I need to find the order of the group generated by the matrices $$\begin{pmatrix}0&1\\-1&0\end{pmatrix},\begin{pmatrix}0&i\\-i&0\end{pmatrix}$$ under multiplication. ...
1
vote
1answer
126 views

$G$ soluble and Unique minimal normal subgroup

Let $G$ be a soluble group and $N$ is only minimal normal subgroup of $G$. Is this $N=C‎_{G}‎(N)$ true?
0
votes
1answer
82 views

The symmetric group $S_4$ and cycles as product of permutations

i have a question, i think it is very easy but i didn't see it: I know that: $$S_4=\langle(1~~2),(2~~3),(3~~4)\rangle$$ Now i want to write $(1~~2~~3)$ and $(1~~2~~3~~4)$ as products of them. ...
0
votes
2answers
158 views

How to find $Z(X)$?

Let $G=\langle a,b : a^{8}=b^2=1, bab=a^{5}\rangle$ and $A=\{I_{G},\phi_{a}\}.$ I can't understand why $\phi_{a}(b)=a^{4}b$. Let ...
2
votes
3answers
77 views

About isomorphism of 2 groups

I want to show that the given $2$ groups are isomorphic to each other; that is, I want to show that: $$G=\langle x,y\mid x^4=1,x^2=y^2,y^{-1}xy=x^{-1}\rangle$$ is isomorphic to the quaternion group ...
3
votes
1answer
45 views

Trace of the action of the Hecke algebra

Let $G$ be any finite group, $H$ a subgroup of $G$, and $\mathcal{R}$ the Hecke algebra associated to this data (i.e. the space of $H$-bi-invariant maps $G \longrightarrow \mathbb{C}$ with the ...
4
votes
1answer
308 views

Can a group of order $55$ have exactly $20$ elements of order $11$? (Clarification)

After I asked this question, which I now understand, I came across a similar question. But I don't understand the answer that was chosen; particularly the part about $34$ elements of order $5$ (but ...
6
votes
1answer
98 views

Isomorphism of complements in semi-direct products

Suppose $G$ is a finite group with normal subgroups $M,N$ and subgroups $H,K$ such that $M \cong N$, $MH=NK=G$, and $M \cap H = N \cap K = 1$. Is it the case that $H \cong K$? Clearly $H \cong G/M$ ...
6
votes
1answer
93 views

Normal products and radicals in finite groups

If $G$ is a finite group with normal subgroups $M$ and $N$, then $MN$ is a subgroup, called the normal product of $M$ and $N$. If $\mathcal{F}$ is a set of finite groups closed under isomorphism and ...
2
votes
0answers
96 views

Schmidt group and maximal subgroups

Let $G$ be a Schmidt group, a minimal non-nilpotent group, so that $G$ is not nilpotent, but every proper subgroup of $G$ is nilpotent. I want to prove $G$ has precisely two classes of maximal ...
3
votes
1answer
69 views

Generalizing the central series to ordinal length

One can generalize the ascending and descending central series by transfinite induction, setting $G _{\alpha +1}=[G_\alpha, G]$ and $ G_\beta= \cap _{\alpha <\beta} G_\alpha $ (and analogously for ...
3
votes
1answer
99 views

Does the class of soluble groups whose $p$-length is $\leq 1$ for all $p$ form a saturated Fitting formation?

Let $\mathcal{F}$ be the class of all soluble groups $G$ such that the $p$-length $l_{p}(G) \leq 1$ for all primes $p$. How do I show ‎$\mathcal{F}$ is a saturated Fitting formation?
1
vote
3answers
106 views

Group Theory Questions-Frattini Subgroup

Given a $p$-group $G$ , and its Frattini subgroup $\Phi(G)$ . How can one prove the following properties: 1) If $H\triangleleft G$ , then $\Phi(G/H)= \Phi(G)/H$ 2) If $G$ is of rank $r$ , and ...
0
votes
2answers
190 views

On techniques of using Sylow Theorems to show that groups of certain orders are not simple

As seen in this answer, a group of order 144 is not simple. Now, I understand the main part of the answer, i.e. where it is concluded that, upon deducing that $n_3 = 16$, it is forced that $n_2 = 1$, ...
2
votes
2answers
217 views

Determining whether or not a group has an element of a specific order

If $|G| = 55$, must it have an element of order $5$ and/or $11$? I'm not quite sure how to determine this. I know it could be possible by Lagrange's Theorem, but I'm stuck otherwise. Any help would ...
5
votes
3answers
302 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
2
votes
2answers
112 views

Proving those elliptic matrices in $\operatorname{SL}_2(ℤ)$ are not conjugate

Set $\mathbf{\Gamma} = \operatorname{SL}_2(ℤ)$, let $\mathbf{H}$ denote the upper half plane. and let $$\Gamma_0 (N) = \left\{ \left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right] ...
2
votes
2answers
99 views

Minimal subgroups lie in the center so group is nilpotent

Let $G$ be a group of odd order. If every minimal subgroup lies in the center, prove that $G$ is nilpotent . Thanks!
3
votes
1answer
226 views

The Fitting subgroup centralizes minimal normal subgroups in finite groups

Let $G$ be a finite group: If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$. Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and $F(G)$ denotes the Fitting subgroup of $G$.
0
votes
3answers
101 views

Proof that a normal subgroup $N$ of $G$ is the identity coset in the group of cosets of $N$

I am not sure how to prove the following: Let $N$ be a group. Prove that for any $n\in N, nN=N$. (Or maybe the following, but I'm not sure it's correct: $n\in N$ $\Leftrightarrow nN = N$) I haven't ...