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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18 views

Probability and Data Integrity

This question is about probability and Security (i.e. data integrity). The scenario I am going to explain is a client-server case where the server may modify the client's data. We define a field ...
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0answers
3 views

Lie algebra of normalizer subgroup

I'm struggling with following problem from book by Olver "equivalence, invariance and symmetry". $G$ is Lie group with Lie algebra $\mathfrak g$ and $H$ its Lie subgroup with Lie algebra $\mathfrak h ...
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0answers
5 views

Chernikov groups are closed under extension

i) A group $G$ is said to be Chernikov if its finite extension of abelian group satisfy minimal condition on subgroups. ii) soluble groups satisfy Min are precisly Chernikov groups. How can ...
-1
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2answers
10 views

Total number of possible binary operations .

If there are n elements in a set the number of binary operations that can be defined are 2n, am I right or wrong ?
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1answer
13 views

About $ S $-free group and normal subgroup of $ S $

Let $ S $ be a group. A group $ G $ is called $ S $-free if no quotient group of any subgroup of $ G $ is isomorphic to $ S $. Let $ G $ is finite group that is $ S $-free. if $ N \lhd S $, then is $ ...
2
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0answers
20 views

On a class of groups of order $p^2q$

Let $|G|=p^2q$ with following conditions: Sylow-$p$ subgroup is normal and is $\langle x,y\rangle \cong \mathbb{Z}_p\times \mathbb{Z}_p$. Sylow-$q$ subgroup is $\langle z\rangle$, and is not normal. ...
-3
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0answers
12 views

coverse of lagranges theorem and example for clt number exsisting group other than A4. [on hold]

Give some simple way to identifying the coverse of lagranges theorem.
0
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0answers
17 views

Permutations and generators and matrix representations.

I am considering all possible permutations of 3 elements and I want to construct these using the generating elements (switch 1 and 3) and (switch 1 and 2). We can represent these using the matrices: ...
2
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1answer
44 views

Question About Proving that $\mathbb Z_p^\times$ Is Cyclic

Statement 1: Let $p$ be prime and that $m$ divides $p - 1$. If there exists an element in $\mathbb Z_p^\times$ with order $m$, then the number of elements in $\mathbb Z_p^\times$ with order $m$ ...
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1answer
26 views

How do you prove the left and right side of an identity of a set are equal?

I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right ...
4
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0answers
23 views

If $G$ acts on $A$ faithfully/freely/transitively, what can we say about its action on $\hat A$?

Let $A$ be a finite abelian group and let $G\subset \operatorname{Aut}A$, with $G$ acting from the left. $G$ acts in a natural way (on the left) on $A$'s character group $\hat A$ by $g\chi = \chi\circ ...
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0answers
13 views

Does one Lie subgroup imply the existence of another in this situation?

Let $n,p,r \in {\mathbb Z}$ with $p \geq 2$, $r \geq 2$ and $n = p + r$. Let $G \subseteq SU\left( n\right)$ be a Lie group and let $S \subset G$ be the subset of all elements in $G$ of the form $g = ...
2
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1answer
24 views

Notation for the set of the subgroups of a group?

Given a group $G$, is there a "standard notation" to denote the set of the subgroups of $G$?
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0answers
20 views

$ \Phi(G) = 1 $ or $ \Phi(G) \neq 1 $?

Let $ G $ is a finite group. Suppose $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup of $ G $. Suppose ...
1
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1answer
33 views

About group actions

Let D={$n_{i}$} be a sequence of integers, $n_{i+1}$ is a multiple of $n_{i}$ ($\forall i$) and $n_{i} \to \infty$. Let us consider a group $H(D)\subset \mathbb{Z}_{n_{0}} \times ...
4
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0answers
34 views

Why $ K \cap H $ is a maximal subgroup of $H $?

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
1
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1answer
18 views

property about centralizer of maximal subgroup

How we can show that for group $G$ (finite non-abelian p-group, I don't know which ones are necessary) and $M$ maximal subgroup of $G$. We can have $C_G(M)\le C_G(\Phi(G))\le Z(\Phi(G))$ $\Phi(G)$ ...
6
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0answers
24 views

What is number of group homomorphisms from $D_{12}$ to $D_{18}$?

I am willing to find out the number of group homomorphisms from $D_{12}$ to $D_{18}$ where $D_m:=\langle r_m, f_m: r_m^m=f_m^2=(r_mf_m)^2=e_m \rangle$ is the standrard dihedral group of order $2m$. ...
4
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0answers
17 views

Can there be a Diaconis-Shahshahani Upper Bound Lemma for Compact Groups?

Let $G$ be a finite group and $\nu\in M_p(G)\subset \mathbb{C} G$ a probability measure on $G$ and let $\pi$ be the uniform distribution on $G$. Denote by $d_\rho$ the dimension of a non-trivial ...
1
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0answers
22 views

Some subset is not a block in group action iff a separation property holds, questions on proof and special cases

Let $G$ be a group acting transitiviely on a set $\Omega$. A nonempty subset $\Delta$ of $\Omega$ is called a block for $G$ if for each $x \in G$ either $\Delta^x = \Delta$ or $\Delta^x \cap \Delta = ...
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votes
1answer
28 views

When does conjugating one generator in $F_2$ give an automorphism?

Let $F_2$ be the free group on generators $x,y$. For $w\in F_2$, consider homomorphisms $\alpha_w : F_2\rightarrow F_2$ sending $$\alpha_w : \left\{\begin{array}{rcl} x & \mapsto & x \\ y ...
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0answers
23 views

$K, N < G$ and $|G:N|$ and $|K|$ coprime, then $K<N$. Group action argument?

A common exercise in finite group theory is Suppose $G$ is a finite group with $K < G$, $N \lhd G$. Suppose that $|K|$ and $|G : N|$ are coprime. Then $K < N$. The normal way to attack ...
1
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1answer
21 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
2
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0answers
34 views

Closure of Set of Fractions with Lowest Terms Condition

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
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0answers
22 views

$Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . [on hold]

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
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0answers
16 views

Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
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0answers
15 views

Difference between 1 (usual) and 1 bar of cayley table?

Why we write 1 as 1 bar in cayley table, instead of usual 1.
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2answers
38 views

What is the center of $\mathbb{C}S_3$?

How do I found the center of symmetric group algebra $\mathbb{C}S_3$? and in general $\mathbb{C}S_n$? I did an example on a smaller group algebra: $\mathbb{C}S_2=\{a (1)+b(12) \mid a,b\in ...
0
votes
1answer
13 views

$f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$?

Let $f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then is it true that $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$ ? Since both $f,g$ are non-trivial , I ...
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0answers
39 views

Question about sub-groups. [duplicate]

Let H,K be sub-groups of G (finite order), proof that if (G;H) and (G:K) are relatively Prime G=HK. Any clue?
1
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1answer
31 views

Chosen maximal subject is a subgroup

Let $ G $ is a finite soluble group and $ N $ be a unique minimal normal subgroup of $ G $. Let $ G = TS $ that $ S $ is the fitting subgroup of $ G $ and $ T = N_{G}(H) $ for $ H \leq G $. Suppose $ ...
-2
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0answers
26 views

Abelian Sylow $p$-subgroups of group $G$. [on hold]

Let $G$ be a non-Abelian group. prove that if $G/Z(G)$ is isomorphic to alternating group $A_4$, then every Sylow $p$-subgroup of $G$ is Abelian.
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votes
1answer
33 views

If two elements commute, does each element commutes with the inverse of the other [on hold]

Let $G$ be a group and $u,v \in G$. Is it possible that $uv = vu$ but $u^{-1}v \ne vu^{-1}$?
1
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2answers
31 views

Sylow p-subgroup of order p does not normalize any other Sylow p-subgroup

Let $P_1,P_2$ be distinct Sylow p-subgroups of $G$ with order $p$. Is it generally true that $P_1$ cannot normalize $P_2$? I've seen algebra textbooks use this fact for $p=3,11$ and they quote 'a ...
-5
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0answers
32 views

The order of the derived group [on hold]

Let $G$ be a non-Abelian group. Prove that if $G/Z(G)$ is isomorphic to the dihedral group $D_8$, then $|G′|=4$.
2
votes
1answer
40 views

Subgroups of generalized dihedral groups

A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements. I know that ...
1
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0answers
21 views

Extension of group

Let $ p = 2 $, and Let $ H $ be a subgroup of a direct product of copies of $ S_{3} $. Why $ H $ is an extension of a $ 3 $-group by a $ 2 $-group, and $ H/O_{p^{\prime}}(H) $ is a $ 2 $-group ?
1
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0answers
15 views

Number of sets containing m decomposable permutations of n objects.

Let $P_{m,n} = \{ \sigma_i \in S_n \}$ be a set containing $m$ arbitrary permutations of $n$ objects. Let $Q_{m,n} = \{\sigma_{ij} = \sigma_i^{-1}\sigma_j \mid \sigma_i, \sigma_j \in P_{m,n} \}$ be ...
2
votes
3answers
32 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
1
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1answer
33 views

Find isomorphism between $S_3$ and $GL_2(F_2)$. [duplicate]

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
0
votes
1answer
24 views

Question in line of proof for first isomorphism theorem

Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\ker \left({\phi}\right)$ be the kernel of $\phi$. Then: $\operatorname {Im} \left({\phi}\right) \cong G_1 / \ker ...
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votes
2answers
57 views

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$? [on hold]

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$? How i can check it?
0
votes
1answer
49 views

Actions of a finite group.

I've been playing around with this proof for a while and I can't seem to figure out where to go from here: I have that $M$ is a manifold, $G$ is a finite group (say, of order $n$), and the action of ...
1
vote
2answers
43 views

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$. I am trying to solve this but i need any ideas or hints to start,any help would be interesting.
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0answers
28 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
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0answers
43 views

The standard representation of $SO(n)$

This fact always bothers me. The group $SO(n)$ is defined as a set of $n\times n $ matrices, so this is a representation of the group in the first place. Yet, it seems that this representation has not ...
-5
votes
1answer
31 views

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$ [on hold]

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$ Any ideas for showing this?
2
votes
1answer
19 views

$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
0
votes
2answers
23 views

How to show #Hom$(C_a, G)=\{x\in G: x^a=e\}$?

I am willing to establish that $$\#\text{Hom}(C_a, G)=\#\{x\in G: x^a=e\}$$ where $G$ is finite group of order $n$ and $C_a$ is cyclic group of order $a$. I started like this: By first isomorphism ...
1
vote
4answers
65 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...