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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Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, $...
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30 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let $...
2
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47 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,3\}$ for $g \ne 1$, and stabilizers are t.i. subgroups, then the Sylow $3$-subgroups have maximal class

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly $3$ points. Also assume that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\...
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51 views

Rigidity of conjugacy classes of finite centerless groups

Let $G$ be a finite group and let $C_1,...,C_k$ be conjugacy classes of $G$. We define the following set: $\Sigma$ = $\{(g_1,...,g_k) \in \prod_{i=1}^{k}C_i \hspace{1mm} | \hspace{1mm} \prod_{i=1}^{...
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62 views

Is there an efficient method to decide whether $gnu(n)<n$ , $gnu(n)=n$ or $gnu(n)>n$?

Denote : $gnu(n)$ = number of groups of order $n$ It is much easier to decide whether a natural number $n$ is group-deficient ($gnu(n)<n$) , group-perfect ($gnu(n)=n$) or group-abundant ($gnu(n)&...
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56 views

Faithful irreducible representation of a finite $p$-group

I want to solve the following exercise in the representation theory field: A finite $p$-group $G$ has a faithful irreducible representation over an algebraically closed field whose characteristic is ...
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49 views

Number of constituents in invariant factor decomposition of kernel of homomorphism

Notation. Given a finite abelian group $ G $, the invariant factor decomposition theorem ensures a the existence of $ k_1 \mid \cdots \mid k_n $, all different, such that $ G \simeq \bigoplus_{i=1}^n \...
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65 views

Number of elements of order $5$ in group of order $5\times 13\times 43\times 73$

Let $G$ be a group of order $5\times 13\times 43\times 73$. Find the number of elements of order $5$. Here is what I do: Since $|G| = 5m$ where $(m,5) = 1$, $m = 13\times 43\times 73$, by Sylow's ...
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38 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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34 views

Size of the intersection of a $p$-Sylow subgroup and a normal subgroup.

Assume $G$ is a group whose size is $(p^r)*m$, where $p$ is prime and $p$ doesn't divide $m$. Let $P$ be a $P$-Sylow subgroup of $G$, and $H$ a normal subgroup of $G$. Lagrange's theorem gives us that ...
2
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53 views

Can Someone approve the formula for the number of groups of order $p^2q$

Here https://www2.bc.edu/~reederma/Groups.pdf on page $112$, a table of the number of groups of order $p^2q$ is given. In the explanations, there is a typo ($\frac{q+5}{5}$ instead of $\frac{q+5}{2}$...
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39 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ is maximal normal. Then $|G/M| = p$.

Let $G$ be a transitive permutation group on $\Omega$ which fulfills the following property (P) (P) each nontrivial element fixes no point or exactly $p$ points. for some odd prime $p$. Further ...
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35 views

There is a group such that satisfy the maximal permutizer condition but not satisfy the permutizer condition ?

$ P_{G}(H) = \langle g\in G \ \vert \langle g \rangle H = H \langle g \rangle \rangle $ called the permutizer subgroup $ H $ of $ G $. $ G $ is satisfy the permutizer condition if $ H < P_{G}(H) $ ...
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52 views

Let $G$ act such that each nontrivial element has at most two fixed point and $P := O_p(N)$. Then $G_{\alpha}$ acts fixed point freely on $P$

Let $G$ be a finite permutation group acting nonregular and transitive such that each nontrivial element fixes at most two points. Lemma: (1) If $p$ is odd and divides the order of $G_{\alpha}$, then ...
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50 views

Order of group $G=\langle x,y\,\,|\,\,x^2=y^2=(xy)^3=1\rangle$.

Find the order of group $G=\langle x,y|x^2=y^2=(xy)^3=1\rangle$. I find that $|G|=5$ but this is not possible because $|x|=2$ not divides $|G|=5$. Thanks for your help.
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99 views

Proof that solvable permutation group whose fixed point set is restricted contains regular normal subgroup or Frobenius group on orbits

Let $p$ be a prime. Let $G$ be a solvable, non-regular, transitive permutation group such that some element fixes no point, and each element fixing some point fixes exactly $p$ points. Suppose that ...
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33 views

Solvability of finite factorized groups

Let $G = AB$ be a finite factorized group. That is, $A \leq G$, $B \leq G$ and for each $g \in G$ there are $a \in A$ and $b \in B$ such that $g = ab$. I'm looking for results which guarantee the ...
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55 views

Is $HK$ abelian?

Assume we have two normal abelian subgroups of the finite group $G$ and we call them $H,K$. My question:- Is $HK$ abelian? I have been able to show that $HK = KH$ and also $HK \unlhd G$. and in ...
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61 views

Isomorphism Type of $\mathbb{Z_8}\times\mathbb{Z_6}\times\mathbb{Z_4} /\langle (2,2,2) \rangle$

Determine the isomorphism type of $\mathbb{Z_8}\times\mathbb{Z_6}\times\mathbb{Z_4}/\langle (2,2,2) \rangle$. Give two proofs: one using elementary analysis of orders of elements, and the other using ...
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45 views

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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41 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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46 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 ...
2
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42 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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93 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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52 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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20 views

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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46 views

$ 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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56 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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50 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 \underline{r}>=&...
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81 views

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 Syl(G)}\textbf{N}_G(S)...
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111 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
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35 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 $S$...
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53 views

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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60 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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67 views

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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0answers
69 views

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 $(\mathbb{Z}/n\...
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59 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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34 views

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 ...
2
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0answers
49 views

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 $...
2
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0answers
35 views

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 ...
2
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156 views

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 ...