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)

0
votes
1answer
33 views

$g$ has order $n$, then $<g>=<g^2>=…=<g^{n-1}>$

$g$ has order $n$, then $<g>=<g^2>=...=<g^{n-1}>$ This should be fairly easy but somehow I just couldn't prove it. I only managed to prove the case when $n$ is prime (which ...
0
votes
1answer
16 views

A transformation of F[x]

I am confused on how to get this problem started. I want to say that they are associated, but I am not sure if that is correct. Let $G$ be the subset of $F[x]$, field $F$, consisting of all the ...
0
votes
0answers
14 views

Matrix elements lie in cyclotomic extension $\mathbb{Q}[\xi_{n}]$

I have $\sigma_{d}(M)$ matrix, and I would like to get back the $M$ matrix. $\sigma_{d}$ is an element of the Galois group of the cyclotomic extension of $\mathbb{Q}$: ...
1
vote
1answer
25 views

Groups and Subgroups elements

Let $G$ be a cyclic group order $12$ with $G=\left<a\right>$. Let $H=\left<a^3\right>$. List the elements of $H$ and find the cosets. I am lost as to what the elements of $H$ would be. ...
0
votes
1answer
40 views

a question about abstract algebra(group theory)

(a)What are the fi nite abelian groups of order 100 up to isomorphism? (b). Say G is a fi nite abelian group of order 100 which contains an element of order 20 and no element with larger order. Then ...
2
votes
1answer
50 views

Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$

I have a ring $\mathbf R =(R, +, -, ., 0, 1)$ (note that there is no invers for multiplication, $R$ is not $\mathbb R$, it is any set for the given algebra). How do you prove that the following does ...
1
vote
2answers
29 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
1
vote
1answer
42 views

Basic doubt about cosets

Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space ...
0
votes
1answer
22 views

group of linear functions and metabelian groups

Let $G$ represent the group of linear functions under composition of the form $x \mapsto ax+b$ where $a,b \in \mathbb{Q}$ and $a\neq0$. Is $G$ a metabelian group?
1
vote
1answer
21 views

Normal group and commutator subgroup question

Let $N$ be a normal subgroup of a group $G$. If $x\in G' \cap N$, is it true that $x \in [G,N]$? I tried to write a product of commutators and set it equal to $n=n_1n_2$. Since $N$ is normal in $G$, ...
0
votes
0answers
16 views

What is the Schreier graph of this group/subgroup/generating set?

Let $G=\langle Pa, b\mid a^3 = \mathrm{id}, (ba)^2 = \mathrm{id}\rangle $, $H = \langle b\rangle$ , and $S = \{a, b\}$. I have been trying to figure out what elements are in this group by finding the ...
0
votes
1answer
15 views

If $k|n, k \geq 2$, then $D_{n}$ has a subgroup isomorphic to $D_{k}$

Restatement of question: If $k|n, k \geq 2$, then the group $D_{n}$ has a subgroup isomorphic to the group $D_{k}$. My attempt at proving the result stated: Let us say that $D_{n}= \{1, \sigma, ...
2
votes
1answer
23 views

Getting fiber bundles from short exact sequences

Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E ...
2
votes
1answer
26 views

A nilpotent group has a normal abelian non-central subgroup

I am working on a proof which involves nilpotent groups. In a proof that I have read about it, it says that, if we have a non-abelian nilpotent group $G$, then there is a normal abelian non-central ...
1
vote
0answers
43 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
1
vote
0answers
12 views

Why does a simple coroot $\alpha^{(i)}$ correspond to a Cartan subalgebra element $H^i$?

I read here that a simple coroot $\alpha^{(i)}$ corresponds to a Cartan subalgebra element $H^i$ and don't understand why this should be the case. Roots are the weights of the adjoint ...
0
votes
0answers
22 views

Representation theory and point groups

Hello everyone :) I have a doubt. I have the point group $C_{3v}$, which is the group $$C_{3v}= \lbrace e, C_{3}, C_{3}^{2}, \sigma_{v_{1}}, \sigma_{v_{2}}, \sigma_{v_{3}} \rbrace$$ $C_{3}$ and ...
1
vote
0answers
38 views

About the elements of Dihedral Group.

I have some difficulties finding the elements of Dihedral Group $D_8$. (Note that e.g., The order of $D_8 = 8$) I know The Geometric Approach for defining the $D_8$. But I always tend to like ...
2
votes
1answer
42 views

Central extension of the Discrete Heisenberg group $H_3(\Bbb Z)$

I want to use the Discrete Heisenberg group $(H_3(\Bbb Z),\times)$ as an example for a presentation on central extensions. $H_3(\Bbb Z) = \begin{bmatrix}1&x&z\\0&1&y\\0&0&1 ...
1
vote
0answers
16 views

Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
2
votes
2answers
31 views

Question about working in modulo?

This question is in essence asking for understanding of a step in Fermats theorem done Group style. For any field the nonzero elements form a group under field multiplication. So let us take the ...
0
votes
0answers
16 views

Why is the k-th convolution of $P_S$ is equal to ${P_S^k}$

For a random walk using transpositions on $S_n$, how can it be explained that the k-th convolution of $P_S$ is equal to ${P_S^k}$. They look to be the same intuitively but how can it written ...
4
votes
2answers
26 views

Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
2
votes
3answers
140 views

What is a free group element that is not primitive?

A primitive element of a free group is an element of some basis of the free group. I have seen some recent papers on algorithmic problems concerning primitive elements of free groups, for example, the ...
2
votes
2answers
43 views

Lie Groups and Matrices

I vaguely remember (maybe I am making this up) this. Is there some sort of result about Lie groups (of a certain class) which classifies them as matrix Lie groups? In other words, given a Lie group G, ...
0
votes
3answers
45 views

Group Theory: How do I determine if an element generates a group?

I was asked if the group $(Z_{17} \setminus \{0\}, \cdot)$ is generated by the element $2$. I understand the concept of generating sub-groups in group theory. If I was given a group $G$ and asked to ...
3
votes
1answer
25 views

If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$ (Proof Verification)

Full question: Let H and K be subgroups of a group G. If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$. I constructed a proof by contradiction and I am wondering whether or not it ...
0
votes
0answers
25 views

Groups, Lagrange theorem [on hold]

True or false: If it's true I should give example Else, to prove why: Finite group with subgroup of finite index
0
votes
1answer
32 views

Special question in group theory

If we have a function $\rho:SL(2,\mathbb{Z})\rightarrow GL(d,\mathbb{C})$ where $m\in SL(2,\mathbb{Z})$ and $\mathcal{M}\in GL(d,\mathbb{C})$ and d is given, for example d=3. And we know that ...
3
votes
2answers
48 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
0
votes
0answers
25 views

Nilpotent by finite group contain characteristic subgroup of finite index

Let $G$ be nilpotent by finite group( i.e there exist normal nilpotent subgroup $H$ such that $G/H$ is finite), i want to prove that $G$ contain a nilpotent charactersitc subgroup of finite index.
1
vote
1answer
46 views

Prove that the maximal normal Abelian subgroup $A$ of metabelian group is equal to $C_G$($A$).

Let $G$ be finitely génerated metabelian group, then there existe maximal normal Abelian subgroup $A$ such that $C_G$($A$)=$A$. I want to prove that $C_G$($A$)=$A$.
0
votes
2answers
44 views

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic? [on hold]

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic?. How would you check this?
1
vote
1answer
36 views

Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
2
votes
1answer
37 views

Isomorphism of two non-abelian groups of order $pq$

Let $p$ and $q$ be two primes such that $q\mid p-1$. Suppose $\phi, \varphi$ are two non-trivial homomorphism from $\mathbb{Z}_q$ to $Aut(\mathbb{Z}_p)$. How to define an isomorphism from ...
1
vote
0answers
17 views
+50

Weights in the Dynkin Basis and Eigenvalues of the Cartan Generators for SU(3)?

The Cartan Generators of $SU(3)$ in the three dimensional rep have eigenvalues $(1,-1,0)$ and $\frac{1}{\sqrt{3}} (1,1,-2)$. Therefore we have the weights: $$ (1,\frac{1}{\sqrt{3}}) \quad ...
1
vote
1answer
19 views

Occurrences of trivial representation is equal to dimension of $\{v\in V:\varphi(g)v=v\}$.

Suppose $\varphi\colon G\to GL(V)$ is a complex representation with character $\psi$. If $W=\{v\in V:\varphi(g)v=v,\ \forall g\in G\}$, why is $\dim W=(\psi,\chi_1)$, where $\chi_1$ is the principal ...
3
votes
3answers
41 views

Number of Sylow $p$-subgroups of a direct product of groups

Let $G$ be the group $S_4\times S_3$ . Prove or disprove the following: a $2-$Sylow subgroup of G is normal a $3-$Sylow subgroup of G is normal I've got $|S_4\times S_3|=144$ and the group as not ...
1
vote
1answer
33 views

How to complete this proof of the Orbit-Stabilizer Theorem?

Let $G$ be a group, $X$ a set, and $*$ and action of $G$ on $X.$ Let $x \in X$ and denote by $\operatorname{Orb} \left( x \right)$ the orbit of $x$ and by $\operatorname{Stab} \left( x \right)$ the ...
0
votes
2answers
40 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
1
vote
1answer
22 views

Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
0
votes
1answer
35 views

Show $G/N $ is cyclic

if $G$ is cyclic, and $N$ is normal to $G$, then $G/N$ is cyclic Can anyone give me a git to start this question? Thanks
1
vote
1answer
35 views

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?
0
votes
0answers
52 views

Fundamental group of projective plane with handles

I was told that the fundamental group of the projective plane with g handles is isomorphic to $\langle c_1, \ldots, c_{2g+1} | c_1^2 \cdot \ldots \cdot c_{2g+1}^2\rangle$. How can I show it? I can ...
0
votes
0answers
59 views

Is this an action of $S_{n}$ on $\mathbb{R}_{n}$?

I am trying to prove that $S_{n}$ acts on $\mathbb{R}_{n}$ with the map $$* : S_{n} \rightarrow \mathbb{R}_{n}, \quad * \left( \sigma, \left( r_{1}, r_{2}, \dots, r_{n} \right) \right) = \left( ...
2
votes
2answers
35 views

Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
2
votes
1answer
36 views

Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
2
votes
1answer
37 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
0
votes
1answer
55 views

Let $g$ be an element of the group $G$. If $|g|>1$ and $|G| = 3*5*7$ is it true that $|g|=3$, $5$ or $7$?

Let $g$ be an element of the group $G$. If $|g|>1$ and $|G| = 3*5*7$ is it true that $|g|=3$, $5$ or $7$? I think that answer is yes, because $|g| \ |\ |G|$. But on the other hand, I don't know ...
0
votes
1answer
37 views

Is it possible to bound recurrence functions for primes?

Would it be possible to bound this function for primes in terms of the maximum difference between the images of the function and their closest primes (for instance, the fifth term is 33 and has ...