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.

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2
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1answer
31 views

Number of elements in Hom$(S_n,\mathbb{C})$

Hox can I determine the number of elements in Hom$(S_n,\mathbb{C})$ for $ n\geq 1$? I thought maybe I can use the thesis that for a normal subgroup $N\subset G$, and a subgroup $H\subset G$, there ...
-2
votes
1answer
32 views

Properties of homomorphisms

I have some problems in how to prove these: Let $f$ be homomorphism from group $G$ to a group $N$. Prove the following: $k\le G$ iff $f[k]\le N$ $f$ is onto iff range of $f =N$ $f$ is one-to-one ...
1
vote
1answer
23 views

Commutator and upper-lower centers question

Let $H$ be a normal group of a group $G$. $H$ is a subgroup of the $k$-th lower center $\gamma_k(G)$. I have a relation like the following $$ [H,G,G,\dots, G] = 1 \qquad (n\; \text{times} \; G) $$ but ...
1
vote
0answers
19 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
2
votes
1answer
58 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
0
votes
1answer
50 views

How many non-isomorphic groups of order 122 are there?

How many non-isomorphic groups of order 122 are there? Let $G$ be a group of order 122.No of Sylow 61 subgroups of order 61=1 and hence it is normal say it is $H$. No. of Sylow 2 subgroups of order ...
3
votes
1answer
58 views

Is there a group homomorphism $f:G\longrightarrow G$ for which $G/\operatorname{Im} f \not\cong\operatorname{Ker} f$? $G$ is finite

Can you find a counterexample to the claim that for all group homomorphisms $f:G \longrightarrow G$, $G/\operatorname{Im} f \cong \operatorname{Ker} f$. Let $G = \mathbb{Z}$; $f(n) = 2n$ is a ...
2
votes
3answers
232 views

Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$

Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $Z_{2p}$ or $D_p$ . Gallian gives a proof as follows : They prove that G = $\langle a \rangle ...
1
vote
1answer
34 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
-2
votes
1answer
54 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
3
votes
1answer
24 views

Generating Constrained Random Distributions

I am trying to help another StackExchange user. We are attempting to fill a 6x6 matrix with 12 A's, 12 B's, and 12 C's subject to the constraint that each row contains 2 A, 2B and 2 C and each column ...
4
votes
1answer
91 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
-1
votes
0answers
30 views

Prove that if $G/Z(G)$ is isomorphic to $\mathbb{Z}_3 ×\mathbb{Z}_3$, then $G$ is isoclinic to an extraspecial group of order $27$. [on hold]

Let $G$ be a non-abelian group. Prove that if $G/Z(G)$ is isomorphic to $\mathbb{Z}_3 ×\mathbb{Z}_3$, then $G$ is isoclinic to an extraspecial group of order $27$.
2
votes
0answers
51 views

Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
0
votes
0answers
26 views

Principal congruence subgroup index in $SL(2,\mathbb{Z})$

Why has the principal congruence subgroup, \begin{equation} \Gamma(N)~=~\Bigg\{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL(2,\mathbb{Z})~|~a\equiv d\equiv 1 ~\text{és}~ b\equiv c\equiv ...
0
votes
0answers
42 views

transitivity of induction

I want to prove the "transitivity of induction" property: Let $ H\leq K\leq G $ where $ G $ is finite. Let M be an $ FH $-module, where $ F $ is any field. Then $ (M^K)^G\simeq^{FG} M^G $. Would you ...
1
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1answer
43 views
2
votes
0answers
39 views

Product of Conjugacy Classes in a Group

Let $G$ be a non-abelian group, and consider $x,y$ in $G-Z(G)$. Let $C(x)=x^G$ and $C(y)=y^G$ denote the conjugacy classes of $x$ and $y$ respectively. Question: What conditions on $x,y$ imply that ...
0
votes
1answer
16 views

Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...
1
vote
1answer
379 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
5
votes
1answer
54 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
-3
votes
1answer
39 views

Generalization of Burnside theorem [on hold]

I probe for understanding the proof of Burnside $ p-q $ theorem: a finite group with a nilpotent subgroup of prime-power index is soluble.
0
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1answer
17 views

Properties of generalized characters

I search for generalized characters which are not characters.Also I want to know that why every generalized character is a difference of characters.
2
votes
2answers
57 views

How do i proof that that the map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is homomorpism?

I'm trying to proof that a map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is a homomorphism but i can't exactly define which is the function to show that. The map $ \varphi $ is a map with for any ...
11
votes
3answers
726 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
1
vote
1answer
38 views

If $X \subseteq G$, $\langle X^{G}\rangle$ is normal in $G$

Let $X^{G} = \{gxg^{-1}: g \in G, x \in X\}$, and define $$\langle X^{G}\rangle = \bigcap_{H \in A} H,$$ where $A=\{H \leq G: X^{G} \subseteq H\}$. We wish to show that $\langle X^{G}\rangle$ is ...
0
votes
1answer
35 views

Frobenius Groups [on hold]

Two questions about Frobenius froups: Let $ G $ be a Frobenius group with kernel $ N $ then help me to prove 1- $ |G:N| $ divides $ |N|-1 $, and 2- If $ L\lhd G $then either $ L\leq N $ or $ N\leq ...
8
votes
1answer
96 views

Eigenvalues of Group Elements and Quaternions

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
0
votes
0answers
32 views

Character of transitive finite permutation groups [on hold]

Let $ G $ be a transitive finite permutation group with permutation character $ \pi $ and let $ \chi $ be an irreducible $ \mathbb{C} $-character. I want to know why the degree of $ \chi $ is at least ...
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vote
0answers
44 views

$ \mathbb{C} $-characters of $ A_5 $

How can we find all of five irreducible $ \mathbb{C} $-characters of $ A_5 $? Precisely how can we construct the character table with the aid of orthogonality relations?
3
votes
2answers
310 views

Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
1
vote
4answers
56 views

Prove that $H$ is normal subgroup of $G$

I have a following question. Let $p$ be a prime and let $G$ be a group and $H$ be a subgroup of $G$. $$ G = \left\{ \begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix} : a,b \in \mathbb{Z}_p, a ...
6
votes
1answer
59 views

Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
7
votes
4answers
89 views

If $G$ is cyclic then $G/H$ is cyclic?

If $G$ is cyclic, then $G/H$ is cyclic? The proof I got goes like this: $G$ is cyclic, so $G=<g>$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So ...
2
votes
0answers
76 views

$G/Z(G) \simeq S_3 \implies |G'|=3$ [on hold]

Let $G$ be a non-abelian group. Show that if $G/Z(G)$ is isomorphic to the symmetric group $S_3$, then $|G'|=3$. Where $$G' = \langle xyx^{-1}y^{-1} : x,y \in G \rangle$$
1
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1answer
64 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 ...
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2answers
30 views

$(h,k).(1,1)=(h\varphi(k)(1),k(1))$ what is $\varphi(k)$?

I am working with: Let $H$ and $K$ be groups and let $\operatorname{Aut}(H)$ be the group of automorphisms of $H$ (under function composition). Suppose also that we are given a homomorphism $\varphi: ...
0
votes
1answer
13 views

If $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ show that $N\lhd G$

Let $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ I need to prove that $N\lhd G$ Attempt: $N\lhd G \iff gng^{-1}\in N$ and it is in $N$ for all $g\in G, n\in N$
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2answers
35 views

If $φ:K → Aut(H)$ is the trivial homomorphism—then the semidirect product is just the ordinary direct product of groups.

I'm working on the question Show that if $φ:K → Aut(H)$ is the trivial homomorphism—i.e., $φ(k) = 1_H$ for every $k ∈ K$— then the semidirect product is just the ordinary direct product of groups. I ...
0
votes
3answers
42 views

Show that $\alpha: G \rightarrow G ,\alpha(g)=g^2$ isomorphism

Let $G$ be a group and $\alpha: G \rightarrow G ,\alpha(g)=g^2$ ,$G$ is abelian group, $|G|=2n+1$ show that $\alpha$ isomorphism Attempt: $\alpha (g_1g_2)=(g_1g_2)(g_1g_2)=(g_1g_1)(g_2g_2)$ ...
2
votes
2answers
39 views

Show that $\psi_a:G\rightarrow G',a\in G,\psi_a(g)=aga^{-1}$ is homomorphism one to one and onto [duplicate]

Let $G$ be a group and $\psi_a:G\rightarrow G',a\in G,\psi_a(g)=aga^{-1}$, I need to show that $\psi_a$ is homomorphism one to one and onto It's not the same question like "Is the conjugation ...
1
vote
1answer
36 views

Facts on $ \mathbb{C} $-characters

My assumption: $ G $ is a finite group & $ \chi $ is a faithful $ \mathbb{C} $-character of $ G $ with degree $ n $ and $ r $ is the number of distinct values assumed by $ \chi $. Now is it true ...
0
votes
1answer
26 views

What is the Lie group that leaves this matrix invariant?

What is the group that leaves \begin{equation} Y = \bigoplus_{j=1}^{k} \alpha_j \mathbb{I}_{2n_j\times 2n_j} \end{equation} invariant under congruence ($Y = XYX^T$) where $\alpha_j \in \mathbb{R}$ ...
0
votes
1answer
45 views

Irreducible $ \mathbb{C} $-character of a finite groups [on hold]

Let $ \chi $ be an irreducible $ \mathbb{C} $-character of a finite group $ G $ and let $ K $ denote the kernel of the associated representation. If $ \chi $ has degree $ n $, is it true that $ ...
3
votes
1answer
40 views

Isomorphism between two groups of order $p^6$

Let $\mathbb{F}_{p^n}$ denote the finite field of order $p^n$. Let $G$ be the group $$ \begin{Bmatrix} \begin{bmatrix} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1\end{bmatrix}\colon ...
1
vote
3answers
51 views

Finite field definition question

My textbook says that $ (\mathbb{Z}_{m},+,*)$ is a field if and only if m is a prime number. However, on Wikipedia it says: "Finite fields only exist when the order (size) is a prime power $p^{k}$ ...
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votes
1answer
70 views

$ \mathbb{C} $-character table of $ D_{14} $ [on hold]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?
4
votes
1answer
154 views

Is $K_n\times K_n$ a Cayley graph?

Let $K_n\times K_n$ be the Cartesian product of two complete graphs. Is $K_n\times K_n$ is Cayley graph or not? I know that I have to use this lemma: A connected graph $G$ is Cayley if and only ...
1
vote
1answer
63 views

Cayley graph of an amalgam

I was wondering what the Cayley graph of an amalgam $A*B$ (over $\{ 1\}$ ) is? $A, B$ are finitely generated. Can't work it out.
10
votes
1answer
602 views

Petersen graph is not a Cayley graph

How can I show that the Petersen graph is not a Cayley graph? I don't know very much about Cayley graphs, I know that they are vertex-transitive, but so is the Petersen graph. It probably has to do ...