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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What are $S_{n}$ and $A_{n}$ in group theory?

What are $S_{n}$ and $A_{n}$ in group theory, and is $[S_{4},A_{4}]=4$? I know that $S$ has to do with permutations, but I am not sure if thats right. Thanks,
2
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2answers
41 views

Why is Rationals w.r.t addition not an Isomorphism to Rationals w.r.t. multiplication?

Question states: Recall the additive groups Z,Q and R, and the multiplicative groups Q* and R* of non-zero numbers. show that: (b) Q is not isomorphic to Q* (c) R is not isomorphic to R* I can see ...
4
votes
3answers
63 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ dots, in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon. Intersection of the lines ...
2
votes
2answers
61 views

Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.

Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an ...
0
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0answers
66 views

verify that the set $\{0,1,2,3\}$ is not a group under multiplication modulo $4$

Given the set $\{0,1,2,3\}$: -Associativity holds for this set -Closure holds for this set (constructing the Cayley table, all entries in the tables are in this set). -there is an identity element ...
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1answer
36 views

Determine if $(((13)),\circ)$ is a normal subgroup of $(S_{3},\circ)$ [on hold]

Let $((13))$ denote the group generated by $(13)$. Is $(((13)),\circ)$ a normal subgroup of $(S_{3},\circ)$? Also is $(((123)),\circ)$ a normal subgroup of $(S_{3},\circ)$? I have just started ...
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1answer
36 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
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1answer
30 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
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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 ...
1
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1answer
22 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
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1answer
29 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 ...
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1answer
52 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
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1answer
22 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 ...
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0answers
29 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 $27$.
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0answers
25 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
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0answers
38 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 ...
2
votes
1answer
30 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 ...
1
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1answer
40 views

Let $G$ be a non-abelian group. Prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 × \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$ [on hold]

prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 \times \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$.
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0answers
47 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 ...
2
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0answers
35 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 ...
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1answer
19 views

If G = pqr where p,q,r are prime, and all the Sylow groups are normal, then is G is abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ where $p,q$ and $r$ are distinct primes with $p<q<r$. If all the Sylow subgroups are normal, then is $G$ abelian? Thank you in advance,
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1answer
89 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 ...
0
votes
1answer
15 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 ...
5
votes
1answer
53 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 ...
0
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1answer
14 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.
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1answer
33 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
49 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 ...
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2answers
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Is group $G$ must abelian, when some condition is given by using exact sequence?

Suppose following exact seqence of groups (E): $1 \rightarrow A \rightarrow G \rightarrow Q \rightarrow 1$ $(A$:abelian normal subgroup of $G$) is given. If $G$ is an abelian group, then $Q$ is ...
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1answer
31 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 ...
3
votes
1answer
55 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 ...
0
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0answers
30 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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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 ...
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votes
4answers
86 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 ...
6
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1answer
56 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 ...
1
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1answer
37 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 ...
11
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3answers
723 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 ...
2
votes
0answers
75 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 $$G' = \langle xyx^{-1}y^{-1} : x,y \in G \rangle$$
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2answers
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$(h,k).(1,1)=(h\varphi(k)(1),k(1))$ 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: ...
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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 ...
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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: $\alpha (g_1g_2)=(g_1g_2)(g_1g_2)=(g_1g_1)(g_2g_2)$ ...
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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 ...
0
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1answer
40 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 $ ...
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0answers
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$ \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
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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
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1answer
31 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 ...
1
vote
1answer
26 views

Homomorphism on U(36)

Question Suppose that $f$ is homomorphism of $U(36)$, $\ker(f) = \{1,13,25\}$, and $f(5) =17$. Determine all the elements that map to 17. What I've tried so far So I've determined that $U(36) = ...
0
votes
1answer
35 views

Is $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{n},n\in \mathbb{N},n>1$ homomorphism?

Let $G$ be abelian group and $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{n},n\in \mathbb{N},n>1$ I need to find if $\varphi$ is homomorphism, and if so to find $ker(\varphi)$ and to tell if ...
5
votes
4answers
97 views

Is $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{-1}$ homomorphism?

Let $G$ be a group and $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{-1}$ I need to find if $\varphi$ is homomorphism, and if so to find $ker(\varphi)$ and to tell if $\varphi$ is one-to-one and\or ...
0
votes
0answers
66 views

Infinite Non Abelian 3- Group [on hold]

Does there exist a infinite non Abelian Group whose every non identity element has order 3.