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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Finding the cycle decomposition of a given permutation.

I am given the following permutation and need to find its cycle decomposition: $\left(\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 5 & 8 ...
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52 views

Let $ G $ be a solvable primitive permutation group. Why the degree of $ G $ is a prime power

Let $ G $ be a solvable primitive permutation group. Why the degree of $ G $ is a prime power and $ G $ has a unique minimal normal subgroup? (8B.4 problem of Finite group theory by Issac) Is ...
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40 views

Finite group in which two Sylow-$p$ subgroups intersect non-trivially but their centers do not

In a non-nilpotent group, some Sylow subgroup is not normal. Suppose $P_1$ and $P_2$ are two Sylow-$p$ subgroups. They may intersect, and I think, this intersection of Sylow-$p$ subgroups has been ...
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45 views

Alternative solution to group theory problem

Exercise Let $G$ be a $p$-group and $g \neq H \leq G$. Prove $\exists g\in G\setminus H$ such that $$gHg^{-1}=H$$ I saw a solution to these exercise using induction on $n$ where $p^n$ is the order of ...
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39 views

If every sylow subgroup of $ H $ is cyclic, is $ H $ soluble group ?

Let $ G $ is a finite group and $ H \unlhd G $, such that $ G/H $ is supersoluble. If every sylow subgroup of $ H $ is cyclic, is $ H $ soluble group ?
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86 views

Proof that $A_n$ is simple for $n \ge 5$, is the one presented here overcomplicated?

In the book Permutation Groups by Dixon & Mortimer, page 78, the well-known fact that for $|\Omega| \ge 5$ the alternating group $Alt(\Omega)$ is simple is proven. It uses a Theorem that if a ...
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51 views

$ N $ be a minimal normal and $ K $ is a $ p $-nilpotent normal subgroup, then $ [ N , K ] = 1 $

Let $ G $ is a finite group and $ N $ be a minimal normal subgroup of $ G $ whose order is divisible by $ p $, that $ p $ is a prime. Prove that if $ K $ is normal $ p $-nilpotent subgroup of $ G $, ...
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Inequality for finite groups?

Let G be a finite group of order |G| and let cl(G) denote the number of conjugacy classes of G . Consider the class of groups which satisfy the inequality: $\qquad$ $3^{cl(G)}$ > $|G|$ $\,$. ...
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37 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 ...
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29 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
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$ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.

Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $. For proof i employ the ...
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75 views

I have a finite permutation group and access to a computer algebra system, how can I recognize the structure of the group?

I have a fixed permutation group $G$, and cannot tell which finite group it really is in a "human readable" way. I also have GAP. Is there a step by step computation to give me the structure of this ...
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51 views

p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
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48 views

Why sylow $ p $-subgroups of $ H $ are Sylow $ p $-subgroups of $ G $?

Let $ H $ is a subgroup of $ G $ that $ \vert G : H \vert $ is a $ \pi $-number and there exist a nilpotent subgroup $ K $ of $ G $ that $ G = HK $.then we can let $ K = K_{\pi}K_{\pi^{\prime}}$, that ...
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34 views

What is Relator Matrix

At the third section of a paper "Computing second cohomology of finite groups with trivial coefficients" by G. Ellis et al., the authors write Suppose that $<\underline{x}\mid ...
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Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
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34 views

Let $ F = VN $ that $ V \cap N = 1 $ . Let $ L = N_{G}(V) $. $ (\vert N \vert , \vert F/N \vert) = $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
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23 views

Find the intertwiner of two equivalent group representations

Let $G$ be a finite group. Let $$\{A(g)\mid g \in G\} \text{ and }\ \ \ \{B(g) \mid g \in G\}$$ be two equivalent unitary matrix representations of $G$. How do I find a unitary intertwining matrix ...
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Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
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59 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
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Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
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60 views

Short exact sequence of groups

Let $1\to N\xrightarrow{\text{i}} G\to H\to 1$ be a short exact sequence of groups. Let $f:G\to N$ be a group homomorphism such that $fi: N\to N$ is an isomorphism. What are some "good" things we can ...
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32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
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Need help in understanding the question. Elemntary number theory

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that ...
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57 views

Showing the existence of $O^{\pi}(G)$

Assume G is a finite group. I am trying to show the existence of $O^{\pi}(G)$, the unique normal subgroup of G minimal such that $G/O^{\pi}(G)$ is a $\pi$-group, i.e. a group whose order is divisible ...
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32 views

How to find the power of generator defined over finite field , $\mathbb F_{2 ^m}$?

List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ...
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If $C_A(F(A)) \le F(A)$ and $C_B(F(B)) \le F(B)$, then this also holds for $AB$ if $A,B \unlhd G$.

Let $A, B \unlhd G$ be normal subgroups of a finite group $G$ such that $$ C_A(F(A)) \le F(A) ~\mbox{ and }~ C_B(F(B)) \le F(B). $$ where $F(G)$ denotes the Fitting Subgroup of $G$. I want to show ...
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does minimality condition imply normal p-sylow subgroup >

Assume that $G$ is a finite group, and $p$ is a prime number dividing the order of $G$. Let $\cal C=\cal C(H)$ be the following condition : "$H$ is a normal subgroup of $G$ and $|G/H|$ is coprime to ...
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Relations between the commutator of two subsets and set theoretical notions

If $X,Y$ are two subsets of some group $G$, then $$ [X,Y] := \langle [x,y] : x \in X, y \in Y \rangle $$ is the commutator subgroup generated by $X$ and $Y$ (where $[x,y] := x^{-1}y^{-1}xy)$. Are ...
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Matrix Elements of Real Represententations

Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \hom V$. Suppose we fix a basis $\{e_i\}_{i\geq 1}$ of $V$ and with respect to this basis we have ...
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Group presentation: How can we determine the group with the presentation below?

Please how can we determine the finite group whose presentation is given as: $$G:=\left\langle g,h|g^4=h^4=1,hg=g^{-1}h \right\rangle?$$
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Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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Adding relators and normal closure

I learned the notion of group presentation. By definition, a group described by a presentation is the quotient of some free group by the normal closure of the relators. In general, given a group ...
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51 views

Sylow 5-subgroups of groups of order $2^n5^m$ are normal

My textbook says: Show that a group of order $2^n5^m, m, n \ge 1$ has a normal 5-Sylow subgroup. I've been banging my head against this problem for days with no success, how can I prove this?
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group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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123 views

Capable group of order 32

A group that can be written as $G/Z(G)$ for some group $G$ is called capable. Can someone list the capable groups of order 32?
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A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in ...
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What is the most mundane & intuitive meaning of the radical of a finite group?

I am asking this question because I am trying to solve this problem from a class note: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G)).$ The note's ...
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Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
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Characterize all $\mathbb Z[i]$ modules of order $21$ and $65$

I am trying to figure out how many $\mathbb Z[i]$ modules (up to isomorphisms) are of $21$ and $65$ elements. I've done a few similar exercises for finite abelian groups ($\mathbb Z$ modules) but I am ...
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Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
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If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
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How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
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Can a chain subcomplex be a direct summand but it's 'origin' is not? (group homology)

Let $H\!\leq\!G$ be finite groups and $C_\ast(H)\!\leq\!C_\ast(G)$ their bar complexes (each $C_k(G)$ is a free $R$-module with basis $(G\!\setminus\!\{1\})^k$). Is it possible that $H$ is not a ...
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33 views

Parametrizing a group element

I'm looking at the problem of parametrizing a group element $g \in SL(n,\mathbb{R}).$ I think I understand the concept of parametrization $-$ just a change of coordinates, right? But I don't ...
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55 views

Sufficiently transitive implies alternating sans Enormous Theorem

According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis ...
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63 views

Existence of a finite group union of self normalizing subgroups

Does a finite group G union of self normalizing subgroups such that the intersection of any two of these subgroups is equal to the unit of group G exist? I don't think so, but I can't prove it. Thank ...
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68 views

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that ...
2
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41 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow ...