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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0
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2answers
33 views

Is the set $SL(2, \mathbb F)$ an Abelian group?

For the set $SL(2,\mathbb F)$, where $\mathbb F$ are entries from either $$\mathbb{Q},\mathbb{R},\mathbb{C} \text{ or } \mathbb{Z}_p \text{ (p is prime)}$$ How should I start by checking this matrix ...
1
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1answer
23 views

Complex numbers modulo integers

Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$? Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates ...
1
vote
1answer
34 views

When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
1
vote
1answer
26 views

How is a symmetric group the subgroup of the group of isometries of three-dimensional space?

So I have this question to solve. I've already shown that the group of rotations of a cube is isomorphic to $S_4$. I need to prove that these two groups are not conjugate when considered as subgroups ...
2
votes
1answer
57 views

Group presentation of Integers $\big(\mathbb{Z,+}\big)$

I can't understand how is it possible to represent the group $(\mathbb{Z},+)$ as follows $$\mathbb{Z} = \big<a\big>$$ with only one generator and no relations ? How can there be no relations ...
-5
votes
0answers
56 views

What are $S_{n}$ and $A_{n}$ in group theory? [on hold]

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,
6
votes
2answers
325 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
76 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
68 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
votes
0answers
75 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 ...
-1
votes
1answer
38 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 ...
1
vote
1answer
43 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
-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
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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
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
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1answer
32 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
23 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
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 $27$.
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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
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0answers
41 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
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 ...
1
vote
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 $D_8$.
2
votes
0answers
48 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
votes
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 ...
-1
votes
1answer
27 views

If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
4
votes
1answer
90 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
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 ...
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.
-3
votes
1answer
35 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
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 ...
2
votes
2answers
60 views

Is group $G$ must abelian, when some condition is given by using exact sequence?

Suppose we are given the following exact sequence of groups where $A$ is an abelian normal subgroup of $G$: $$1 \rightarrow A \rightarrow G \rightarrow Q \rightarrow 1\tag{E}$$ If $G$ is Abelian, ...
0
votes
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
56 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
votes
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 ...
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 ...
7
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
57 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
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 ...
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 $$G' = \langle xyx^{-1}y^{-1} : x,y \in G \rangle$$
0
votes
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
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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 $g\in G, n\in N$
0
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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 ...
0
votes
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 $ ...
1
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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
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 ...