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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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0answers
16 views

How to check the property of projective linear group

In Isaac's Finite Group Theory Page 50, it states: A $Sylow$ $2$-subgroup $P$ of $G=PSL(2,7)$ of order 168 is contained in two maximal subgroup of $G$, each of order $24$, and $Z(P)$, which has ...
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1answer
13 views

A question regarding “equality” of word lengths for two minimal generating sets of a finite group

Let $G$ be a finite group $d(G) = \min_{<S>=G}|S|$. Suppose that $|X|=|Y|=d(G)$ and $<X>=<Y>=G$. Let $|g|_X$ be the word length of $g$ with respect to $X$ and $|g|_Y$ be the word ...
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1answer
21 views

Prove that if a subgroup $H$ of a finite cyclic group $G=\langle a \rangle $ of order $n$ is generated by $a^m$, then $m$ is a divisor of $n$.

Prove that if a subgroup $H$ of a finite cyclic group $G=\langle a \rangle $ of order $n$ is generated by $a^m$, then $m$ is a divisor of $n$. It can be easily shown that $m$ is the least positive ...
5
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1answer
50 views

A polynomial algorithm to determine whether a finite group is nilpotent

Does there exist a polynomial (in respect to the order of the group) algorithm that given a Cayley table of a finite group determines, whether a group is nilpotent or not? There do exist polynomial ...
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0answers
36 views

Choice of symbols: $O_p(G)$, $O^p(G)$, and $O_\infty(G)$

For a finite group $G$ and a prime number $p$, several normal subgroups are defined as follows: $O_p(G)$ = the largest normal $p$-subgroup of $G$ ($p$-core) $O^p(G)$ = the smallest normal subgroup $N$...
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0answers
37 views

Show that the center of quaternions group $\textit{Q}$ is generated by the unique element with order 2.

$\textit{Q}$ is a group with order $8$, generated by $a,b$ where $a^4=1$, $b^2=a^2$ and $bab^{-1}=a^{-1}$. I already proved that the unique element of $\textit{Q}$ with order $2$ is $a^2$. How can I ...
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3answers
43 views

ord$(h)|\max\{\text{ord}(g)|g\in G\}$ for all $h\in G$.

Let $G$ be a finite abelian group and $n:=\max\{\text{ord}(g)|g\in G\}$. Now I have to proof that ord$(h)|n$ for all $h\in G$. My idea was: Let $g\in G$ with ord$(g)=m<n$. Then because of the ...
4
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2answers
18 views

Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
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1answer
31 views

Group of the Image of a character of a finite abelian group is cyclic [on hold]

Let $G$ be a finite abelian group and $\hat{G}$ be its dual group. $\gamma$ is a character of $G$, i.e. $\gamma:G\rightarrow \mathbb{C}$, $|\gamma(x)|=1$, $\forall x \in G$ and $\gamma(x+y)=\gamma(x)\...
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0answers
46 views

If $x∊S_n$ normalizes but does not centralize a subgroup of prime order $P$, show that $x$ fixes at most one point in each orbit of P$. [on hold]

Let $P⊆S_n$ be a subgroup of prime order and suppose $x∊S_n$ normalizes but does not centralize $P$. Show that $x$ fixes at most one point in each orbit of $P$.
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0answers
30 views

Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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2answers
32 views

Q question about proving isomorphism of abelian groups

Suppose that $\mathbb{Z}_n^{+}$ denotes the cyclic group of order $n$. Question a: Consider the group $$ G=\mathbb{Z}_{n_1}^{+}\times \mathbb{Z}_{n_2}^{+}\times \ldots \mathbb{Z}_{n_k}^{+} $$ where ...
2
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1answer
26 views

Cayley's theorem: Is $C_5$ isomorphic to $ \langle (1 2 3 4 5) \rangle \leq S_{5}$?

I want to use Cayley's theorem to determine a subgroup in $S_n$ ( for n as small as possible) which is isomorphic to $C_{5}$. I believe this subgroup to be $ \langle (1 2 3 4 5) \rangle $. Here is ...
4
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0answers
49 views

Double centralizer of derived subgroup

Let $G$ be (finite) matabelian group; define $W(G):=C_G(C_G(G'))$; $G'$ is derived subgroup of $G$. If $\mathcal{F}$ is the collection of all maximal abelian normal subgroups of $G$ which contain $G'...
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1answer
36 views

Regarding the Intersection of $p$-Sylow Subgroups

The Problem: Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Prove that a normal $p$-subgroup of $G$ is contained in every $p$-Sylow subgroup of $G$. My Attempt(s): Here's a slight ...
1
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1answer
54 views

How to understand the structure of the interesting graph obtained from the group?

Let $G = A_5$ and $H < G$ is subgroup of $A_5$ generated by $(12)(34)$, and $(125)$. Define graph $\Gamma$ by a vertex set is element of $G$ and elements $x$ and $y$ adjacent if $|H^x \cap H^y| =...
6
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0answers
64 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
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0answers
34 views

Conjugacy Classes and Irreducible Representations

For finite groups number of inequivalent irreducible complex representations equals the number of conjugacy classes. When the group is $S_n$ it is easy to see that both are available one each for ...
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1answer
52 views

Doubt regarding “Elementary approach to proving that a group of order 9 is Abelian”

I am trying to understand the solution of this problem . I am unable to understand why : If $yx=x^2y$, then $yxy^{-1}=x^2$. This means that $y^3xy^{-3}=x^8 $ It seems like I am missing something ...
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0answers
50 views

What if the sum in the RHS of the Class Equation is extended to the whole $G$?

For a finite group $G$ of order $n$, the Class Equation reads: $$n=\sum_{b_j \in B}[G:C_G(b_j)]$$ where $B$ is a set of representatives of the conjugacy classes of $G$. Q: Can else come from ...
2
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3answers
187 views

Prove that every group of order 15 is abelian? [duplicate]

I had seen this proof at many places, but everywhere sylows theorem is used. So is their any way to solve it without using sylows theorem?
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1answer
96 views

Axiomatisable Groups

Let A be a set of sentences (“proper axioms”) in a first-order language L with equality. Let us write Mod(A) for the class of all models of A which respect equality. We say that a class of L-...
2
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1answer
46 views

A dodecahedron out of five tetrahedra AKA Partitioning an orbit in X/K into orbits under cosets of H in K

Consider the orbit space $X/K$ with $X$ a symmetric space and $K$ a group. Let $x$ represent an orbit $Kx$ in $X/K$. Now let's introduce a subgroup $H \subset K$, split up $K$ into cosets $aH$ (with $...
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0answers
39 views

embedding a proper normal subgroup of a p-group

I am trying to prove this claim: Let $G$ be a finite $p$-group and $H$ be a proper normal subgroup of order $p^k$,then $H$ can be embedded into a normal subgroup of order $p^{k+1}$. Here is my ...
2
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2answers
39 views

Inductive Proof of Group with Prime Decomposition is Isomorphic to Direct Product of Cyclic Groups

My lecturer set as a bonus exercise the following induction proof: If $G$ is a finite abelian group $|G| = p_1^{n_1} \cdots p_s^{n_s}$ is the decomposition of $|G|$ into a product of distinct prime ...
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1answer
30 views

Isomorphism of a quotient group and a subgroup [closed]

Let $G_1$ be a finite group with a subgroup $G_2$ and a normal subgroup $H$. Suppose that $G_1/H$ is isomorphic to a subgroup of $G_2$. Do we have $G_1/H$ is isomorphic to $G_2$?
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1answer
30 views

A group of order $p^2q^r , p\lt q$ is not simple [closed]

Prove: A group of order $p^2q^r , p\lt q$ is not simple I only see proofs for $r=1$, how can I deduce for $r\in\mathbb{N}$? Any help would be appreciated.
1
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1answer
39 views

Modulo congruences and remainder

Let $x \in \mathbb{Z}$ Show that if $N\mid M$ then $(x\pmod M)\pmod N = x \pmod N$ My proof: Assume $x \in \mathbb{Z}$ is arbitrary. Then define $x\pmod M =r \iff x \equiv r\pmod M$ where $0\leq r &...
8
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1answer
541 views

Is this a group? If so, what group is it?

I have the following group (at least, I think it's a group) generated by $\langle a,b,c \rangle$ where the operation $\cdot$ obeys the following rules: $a^2=b^2=c^2=1$ (where $1$ is the identity). $\...
1
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1answer
47 views

Construction of a group in Magma

I need to construct the following group in Magma: given $H=(C_2)^3 \rtimes (C_7 \rtimes C_3) \times C_3$ (so $\operatorname{SmallGroup}(168,43)\times C_3$), there is a non-split central extension by $...
1
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1answer
31 views

Find all group of order $20$ which is a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$

Problem: Find all group of order $20$ that are a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$. My attempt: We knew that a cyclic group of order $n$ is isomorphic ...
5
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1answer
85 views

Hall subgroups of $\mathrm{PSL}$

The following is an exercise in Peter Cameron's notes on classical groups. Exercise 2.10 (a) Show that $\mathrm{PSL}(2,5)$ fails to have a Hall subgroup of some admissible order. (b) Show that ...
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2answers
54 views

Prove for a normal subgroup

So my task is to prove that if $N$ is finite group and $G$ is a normal subgroup of $N$ and $(|G| , |N : G|) = 1$ , for every subgroup of N (let's call it F) if $|F|$ divides $|G|$, $F$ is a subgroup ...
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2answers
36 views

Finite Group Property Proof [closed]

I'm trying to prove that for every finite group $G$ there is a $n$ such that $g^{n}=1$ $\forall g \in G$. Any ideas?
2
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1answer
73 views

Divisibility of cubes by 7

Show that if if $x^3+y^3=z^3$ for some $x,y,z$ $ \in \mathbb{Z}$ then one of $x,y,z$ is divisible by $7$. I'm stuck on this problem. I know that for any integer $n\in \mathbb{Z}$, $[n^3 \pmod{7}]$ $\...
2
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0answers
47 views

How to enumerate Von Dyck groups?

I would appreciate an algorithm to list all elements of a given Von Dyck group $D(p,q,r)$, each once, in a format that would allow me to find compositions and inversions within that list. ...
1
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1answer
22 views

Find all semidiect product of $(\mathbb{Z}_4,+)$ by $\text{C}_2 = \{{a\}}$

Problem: Find all semidiect product of $(\mathbb{Z}_4,+)$ by $\text{C}_2 = \{{a\}}$ (the cyclic group of order $2$). My attempt: We know that $(\mathbb{Z}_4,+)$ is a cyclic group of order $4$. To ...
3
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1answer
38 views

Center and derived of the subgroup generated by $\langle(1,2,3),(1,2,4),(5,6)\rangle$ in $S_6$

Given the following subgroup of $S_6$: $G=\langle(1,2,3),(1,2,4),(5,6)\rangle$ I'm asked to show that $H= \langle (1,2,3),(1,2,4)\rangle$ is normal on G and isomorphic to $A_4$ and also to find the ...
2
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1answer
30 views

Cycle types of elements of centralizers of transitive permutation groups

I think the following is true, but I'm not sure how to prove it. Any help would be greatly appreciated. Suppose $G$ is a transitive subgroup of $S_n$ and $C$ is the centralizer of $G$ in $S_n$. If $...
2
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2answers
86 views

Let $G$ be a non-abelian group of order $10$. Prove that G has a trivial center.

I have done this as follows: Let $G$ be a non-abelian group of order $10$. If possible let a non-identity element say $a \in G$ is in $Z(G)$. Now by lagrange's theorem , $|a|= 2,5$ or $10$. If $|a|...
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0answers
25 views

Number of permutations $\sigma\in S_n$ with $\sigma(k)\neq k$ for all $k=1,\ldots,n$ [duplicate]

For the symmetry group $S_n$ ($n\geq1$), how many permutations $\sigma$ exist with the property that $\sigma$ doesn't map any element of $\{1,\ldots,n\}$ to itself? I know I can try to do a counting ...
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0answers
34 views

The construction outer semidirect product from two groups.

Problem: Let $N$ and $H$ be any group and let $\theta \colon H \rightarrow \text{Aut}(N)$ is a homomorphism. The the set $$G = \{(x,h) \mid x \in N, h \in H\},$$ is a group with operation defined as ...
1
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2answers
27 views

Some properties implied from the definition of semidirect product.

Problem: Let $G$ be an inner semidirect product of $N$ by $H$. For each element of $H$, denote $\theta_h \colon N \rightarrow N$ is a map defined by $\theta_h (x) = h x h^{-1}, \forall x \in N$. Prove ...
0
votes
1answer
60 views

Cycle types of elements of normalizers of point stabilizers of transitive permutation groups

Is the following true? Suppose $G$ is a transitive subgroup of $S_n$ and $G_1$ is the point stabilizer of 1 in $G$. Let $N$ denote the normalizer of $G_1$ in $G$. Suppose $\sigma\in N\setminus G_1$ ...
4
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0answers
40 views

Finite groups of cyclicality index 3

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation: $$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
3
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0answers
43 views

How to find the irreducible constituents of a Brauer character with MAGMA?

I'd like to ask the following MAGMA-question: How to find the irreducible constituents of a Brauer character with MAGMA? Let $G$ be a finite group. Moreover, let $k$ be a finite field of ...
0
votes
1answer
23 views

Cardinality of an intersection of two submodules.

Assume $p$ is a prime number and $q = p^2$. Denote by $A$ the ring $\mathbb{Z} / q \mathbb{Z}$. Consider a finite type module $M$ over $A$ whith cardinality $q^N$ where $N$ is an integer, $N>0$. ...
-1
votes
1answer
42 views

Finding a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$ [closed]

Can somebody help me to find a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$? I actually dont even know how to start :/ Thank you!
2
votes
1answer
48 views

For what $n$ is $A_n$ a marginal subgroup of $S_n$?

For what $n$ is $A_n$ a marginal subgroup of $S_n$? Strict definition of marginal subgroups and brief overview of their properties can be found here What have I tried: $A_n$ is characteristic in $...
0
votes
1answer
58 views

Every group of order 440 is solvable

Given a group G of order 440, it has a unique subgroup of order 11 which is normal in G. Let's call it H. H is clearly solvable, if G/H was solvable, so it would be G. However I cannot seem to be ...