# 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.

8,540 questions
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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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 $\gamma(x+y)=\gamma(x)\... 0answers 46 views ### If$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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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-...
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### 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 ...
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### 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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### 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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### 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?
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### 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 ...