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Questions tagged [finite-groups]

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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How do generators of a group work?

$G$ is a group, $H$ is a subgroup of $G$, and $[G:H]$ stands for the index of $H$ in $G$ in the following example: Let $G=S_3$, $H=\left<(1,2)\right>$. Then $[G:H]=3$. I know the definition of ...
-4
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0answers
31 views

Some properties of a group [on hold]

(a) Suppose $G$ is a finite group. The order of an element $g ∈ G$ is defined to be the smallest positive number $n$ such that $g^n = e$. Show that the total number of elements of $G$ of order $> 2$...
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0answers
26 views

Representation Theory Direct Sum Composition

Let $A_5 \subset S_5$ be the alternating group on five letters. (a) Describe, as a direct sum of irreducible representations of $A_5$, the restrictions to $A_5$ of each of the irreducible ...
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0answers
20 views

If H is permutable and A is subgroup of H then A is permutable?

Let $G$ be a finite group and $A\le H$ and H is a subgroup (but doesn't equal) of $G$ . How to show that if $H$ is permutable in $G$ ( i.e $HB = BH$ for all $B\le G$ ) then $A$ is permutable ...
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1answer
39 views

Is it true all centralizer of G are abelian?

Suppose $G$ is a finite group such that $\frac{G}{Z(G)}\cong Z_p\times Z_p\times Z_p$. Is it true all centralizers of G are abelian?
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1answer
44 views

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [on hold]

Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the ...
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0answers
21 views

How to find normalizer to a subgroup of the Pauli group?

The Pauli operators are given by: $X = \left( \begin{array} { c c } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right) , \quad Y = \left( \begin{array} { c c } { 0 } & { - i } \\ { i } &...
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0answers
20 views

On $n$th class-preserving automorphism of finite $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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0answers
21 views

Direct product of a finite subgroup. [on hold]

I'm stuck with the next exercise. Let $G$ be a finite group and $N_1,\dots, N_k$ a normal subgroups of $G$ such that $$|G|=|N_1|\cdots |N_k|$$where $|G|$ is the order of the group. Prove that $G$ ...
2
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2answers
65 views

How to show that $⟨a,b | aba=bab⟩$ is not the trivial group?

I want to show that G = $⟨a,b | aba=bab⟩$ is not the trivial group I tried to find homomorphism $\phi$ from $G$ to $\mathbb Z$ which maps $a$ to $0$ and $b$ to $1$ (or $b$ to $0$ and $a$ to $1$) ...
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0answers
40 views

Symmetry group of cube

I want to carefully trace the entire sequence of interactions between the structures when getting the symmetry group of the cube. I'm just learning how to work with structures and diagrams, so please ...
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0answers
27 views

Is there a general approach for finding the Normalizer of a subgroup? [on hold]

I am currently facing a problem where I need to find the Normalizer of a subgroup. I have found examples where people compute the normalizer for specific subgroups, but no general approach. Does ...
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0answers
20 views

About the differences between definitions of “Capable Group”

I am looking for the properties of groups having "immediate Descendants", in other therm, "Capable Groups"; The problem that I fond is that "Capable Group" could have many meaning! So, could you ...
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0answers
29 views

Generators of $1+\Delta (G)$, where $\Delta(G)$ is augmentation ideal of group ring $FG.$

Let $FG$ be a finite group ring of a finite non abelian $p$-group $G$ over finite field $F.$ It is well known that augmentation ideal $\Delta(G)=J(FG)$ has basis as the set $\{g-1:g\in G, g\ne 1\}$,...
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1answer
52 views

Show that $\phi(g)$ is an even permutation

Let $G$ be a group of order $n$ then $G$ is isomorphic to a subgroup of $S_n$,Denote it by $\phi$, $\phi:G\to S_n$ be an monomorphism of Cayleys Theorem Let $g\in G$ has order $k$.Show that $\phi(...
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0answers
15 views

Convolution of probabilities on finite groups

I was reading a book on group and representation theory and came across the following which I don't understand, I'd appreciate any help. Suppose P and Q are probabilities on a finite group G. Thus $...
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1answer
36 views

Computing quotients of group by elements of its lower exponent$-p$ central series

Let $G$ be a finite $p−$group of number of generators $d$ and exponent$−p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G)=1$ in the series $$ G=P_0(G)≥...≥P_{i−1}(G)≥P_i(G)≥... $$ ...
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0answers
32 views

$Spin_n(q)$, $SO_n(q)$, $\Omega_n(q)$ and their projective images

I am studying about the structure of orthogonal groups and struggling to understand the relations between groups in the title: From algebraic groups point of view, it is known that $Spin_n(q)$ is ...
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0answers
30 views

Presentations of $D_8$ using permutations

I fond a lot of examples using presentation of $D_8$ by generators which are permutations of $S_4$. 1) How many presentations could be found? 2) Could it be presented by permutations of $S_5$ or ...
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0answers
41 views

Monster group poster

I have seen a number of questions here on how to intuitively understand the Monster Group. My questions is, is there a way one can create an image or series of images suitable for putting on a poster ...
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1answer
12 views

Hamiltonian decomposability of 4- regular graphs

If a 4 regular graph is hamiltonian can we say it is hamiltonian decomposable ? Thanks a lot in advance
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1answer
44 views

There are only two groups of order six, up to isomorphism: $\mathbb Z_6$ and $S_3$. [duplicate]

Let $G$ be group with order $6$. Prove that either $G$ and $\Bbb Z_{6}$ are isomorphic binary structure or $G$ and $S_{3}$ are isomorphic binary structure. I know that for isomorphic binary ...
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0answers
49 views

If $G'=G$, and suppose $N_G(P)$ is solvable for $P\subseteq G$, show that $ G$ is simple.

Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $P\subseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple. I ...
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3answers
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Sylow $3$-subgroups of an order $180$ group

My task is to show that if $G$ is a group with order $180 =2^23^25$ with $36$ Sylow $5$-subgroups, then there are two Sylow $3$-subgroups $H$ and $K$ such that $|H \cap K| = 3$. The number of Sylow $...
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0answers
18 views

How to construct the homomorphism in semidirect product of $Z_3$ and $Z_{13}$?

I know that in the semidirect product of $A$ and $B$, the homomorphism $\phi:A\rightarrow Aut(B)$ should be $\phi_y(x) = yxy^{-1}$ but have no idea how to construct one for $\phi:Z_3\rightarrow Aut(Z_{...
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1answer
27 views

Prove the existence of homomorphism.

I am trying to answer the following question. Is there any group homomorphsim $\phi: D_4 \rightarrow S_5$?
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0answers
48 views

A visual subject in group theory

My math teacher gave us the following instruction: « Pick any subject of your choice (in math of course) and in 2 months present it to the class ». I really like this idea, and I really like group ...
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0answers
21 views

Homomorphism and relative order questions

1) I have to prove that if $\exists$ a nontrivial homomorphism $\phi:A\rightarrow B$, where A and B are finite and Abelian, then $|A|$ and $|B|$ are not relatively prime. I know that $\phi(A)$ is a ...
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0answers
29 views

Relation between odd order theorem and the fact that every polynomial of odd order has at least one root

It is known that there is connection between solvability of a group and expressing the roots of a polynomial by radicals, which is something I will study in this semester; however, by the odd order ...
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0answers
17 views

Semi direct product of Quaternion group with cyclic group of order p.

I am interested in knowing the semi-direct product of Quaternion group $Q_8$ with $c_p$, i.e. cyclic group of order $p$ where $p$ is a odd prime. We know that $\text{SL}_{2}(\mathbb{Z}_{3})$ is a ...
2
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1answer
31 views

About qoutients of Lower Exponent$-p$ central Series

Let $G$ be a finite $p-$group of number of generators $d$ and exponent$-p$ class $c$, that is $c$ is the smallest integer satisfying $P_c(G) =1$ in the series $$ G=P_0(G) \geq ...\geq P_{i-1}(G)\geq ...
2
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2answers
49 views

How can I prove that $G/M\cap N \cong M/(M\cap N) \times N/(M\cap N)$ where $M,N \triangleleft G$ and $M.N=G$?

If $M\cap N = \{e\}$ then $M/M\cap N = M$, $N/M\cap N = M$ and $G/M\cap N = G$ and $G = M\times N$ trivially by the def of direct product. If $M\cap N \neq \{e\}$, I was trying to follow the same ...
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1answer
46 views

Trouble understanding semidirect product of $Z_3$ and $Z_{13}$

Here a representation for a non Abelian of order 39 is given Finding presentation of group of order 39 by Pisco. Am I correct in understanding that the non-trivial homomorphism in this case is: $\phi:...
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1answer
16 views

$2$-factorizations of a $4$-regular graph

Suppose there is a $4$ regular graph. Then it has $2$ edge disjoint $2$-regular spanning subgraphs. Let the spanning subgraphs be $T_1$ and $T_2$. Can there be another pair $T_3$ and $T_4$ of edge ...
2
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1answer
28 views

How to prove that semidirect product of $Z_{13}$ and $Z_3$ is non Abelian for a non-trivial homomorphism

The semidirect product of $Z_{13}$ and $Z_3$ is given here Finding presentation of group of order 39 as $\{x,y | x^{13} = y^3 = 1, yxy^{-1} = x^3\}$. I understand how this is arrived at but to show ...
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1answer
26 views

isomorphism of normal subgroup [closed]

I am reading group theory (particularly isomorphism) in the algebra, and stuck on a problem. Hope you guys will help me out: Let $G$ be finite group, and $A$,$B$ be normal subgroups of $G$ such that $...
3
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1answer
37 views

orthogonal group what does it represent

Let $A$ be an finite abelian group and $B$ be a subgroup of $A$. Then we defined the orthogonal of $B$ : $$B^{\perp} = \{f:(A,+) \to (\mathbb{Q}/\mathbb{Z},+) \mid \forall b \in B ,f(b) = 0 \}$$ I ...
2
votes
1answer
63 views

How to find an example of a non Abelian group of arbitrary finite order? eg. $39$

I was thinking of building on top of known non Abelian groups, like $S_3$, and taking a direct product with $\Bbb Z_n$'s but those groups' order would be a multiple of order of $S_3$. So, is there ...
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0answers
29 views

Let $(G, \cdot)$ be a finite group, $H < G$ (proper subgroup), and $x \in H$ such that $C_{H}(x) < C_{G}(x)$. What consequences follow? [closed]

I have a problem where I am given these two facts, but I do not seem to be able to hit upon relevant consequences of them. The one I could figure it out is the most obvious one: $C_{H}(x) = H \cap C_{...
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1answer
47 views

Let $(G, \cdot)$ be a finite group, with $H \triangleleft G$ and $[G:H]$ prime. What are the consequences of this? [on hold]

I have a problem where I am given these two statements as a fact. However, I am having trouble figuring out what the relevant consequences of these statements would be, since I have only been able to ...
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0answers
38 views

If $|G|=p^nm$ where $\text{gcd}(p,m)=1$ and $|G|<m!$, then can $|G|$ be simple?

I think my professor had a type-o on recent homework. He claims that if $G$ is a group and $p$ prime, then if the title's conditions hold we must have $G$ not simple. This doesn't seem correct to me, ...
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votes
3answers
48 views

Exponent of a non abelian group. [closed]

Let $G$ be any finite non-abelian group such that $a^{p^3}=1$ for each element $a\in G$ ,where $p$ is any prime number $\geq 3$. Then can i say that exponent of $G$ must be $p^3$ as any group of ...
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1answer
50 views

Express (1,…,n) as a product of 2-cycles

In Alan F. Beardon's "Algebra and geometry" he asks in an exercise to express $(1\ \ldots\ n)$ as a product of two cycles: Show that $(1\ 2\ 3\ 4)=(1\ 4)(1\ 3)(1\ 2)$. Express $(1\ 2\ 3\ 4\ 5)$ as ...
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0answers
35 views

Conjugacy classes of stabilizer subgroups

Let $G\subseteq GL(V)$ be a complex finite reflection group. I would like to understand the stabilizer subgroups of $G$, and their normalizers. By this I mean (the conjugacy classes of) subgroups $H&...
0
votes
1answer
46 views

Equivalent condition for a group to be cyclic [duplicate]

Group $G$ is cyclic $\iff $ Every subgroup of $G$ is like $G^m = \{g^m\ |\ g\in G\}$ for some $m \in \mathbb N$. $G^m = \{g^m\ |\ g\in G\}$ means $\forall x \in G^m, \exists\, g \in G$ s.t. $x=g^m$....
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vote
1answer
26 views

Let $(G, \cdot)$ be a group, and $H \leqslant G$. Let $x \in H$, what $C_H(x) < C_G(x)$ mean? ($C_A(x)$ notation for “centralizer of $x$ 'in A”)

Let $(G, \cdot)$ be a group. For any $x \in G$, we write: $$ C_G(x) = \{z \in G \mid z \cdot x = x \cdot z\}$$ Let $H \leqslant G$ (subgroup of), and $x \in H$. What does it mean when we write: $$ ...
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votes
2answers
41 views

Which of the following is isomorphic to the group of units of $\mathbb{Z}_{35}$? [on hold]

$C_2 \times C_2 \times C_6$ $C_2 \times C_{12}$ $C_{24}$ The group of units of $\mathbb{Z}_{35}$ is $\mathbb{Z}_{35}^\#= \left\{ 1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,...
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0answers
31 views

Permutations on vertices of cubes and hence finding volume enclosed by the vertices

Denote $C$ to be the cube $C=\{(x_1,x_2,x_3)|0 \leq x_1,x_2,x_3 \leq 1\}$ and let $V=\{ (x_1,x_2,x_3)|x_1,x_2,x_3 \in \{0,1 \} \}$ be the set of vertices of the cube. Let $A=$convex$((0,0,0) , (1,0,...
1
vote
1answer
38 views

Every finite group has a chief series

A chief series in a group $G$ is a series of normal subgroups such that $1=N_0 \triangleleft N_1 \triangleleft ... \triangleleft N_n=G$, for which each factor $N_{i+1} / N_i$ is a minimal (non-...
0
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0answers
26 views

Questions on a powerful generalization.

The initial variation was to prove that for $u,v$ two different prime integers such as $\gcd(u,v)=1$ we have $u^{v-1}+v^{u-1}\equiv 1\pmod{uv}$. I solved this question by using the CRT and Fermat ...