# 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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### $O_{\pi}(G/N)=PN/N$

Is the following statement correct? Let $\pi$ be a set of primes and let $P$ be a Hall $\pi$-subgroup of $G$. If $N$ is a normal subgroup of $G$ such that $G'\subseteq N$, then $O_{\pi}(G/N)=PN/N$. ...
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### For finite groups, $[F(G),E(G)]=1$

Let $G$ be a finite group, $F(G)$ be the Fitting subgroup of $G$ and $E(G)$ be the layer of G (subgroup generated by components of $G$). Then $[F(G),E(G)]=1$. My attempt: Since $E(G)$ is in ...
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### If $EZ/Z$ is a component of $G/Z$, then $E'$ is a component of $G$.

I have a doubt in the proof of the following given in Kurzweil and Stellmacher: Let $Z$ and $E$ be subgroups of a finite group $G$ such that $Z\leq Z(G)$ and $EZ/Z$ is a component of $G/Z$. Then $E'$ ...
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It is well-known that tensor products of irreducible representations of a finite group decompose into direct sums of irreducible representations according to fusion rules $$\Gamma_i \otimes \Gamma_j=\... 0 votes 0 answers 29 views ### Can we extend this definition of graph of group? Let G be finite nonabelian group. According to Aalipour et al. (2016), they defined the enhanced power graph of G as a simple undirected graph where the vertices are all elements of G and two ... 0 votes 2 answers 56 views ### If G=AB and |G|=|A||B|, then G=A^{-1}B. Let A and B be subsets of a finite Abelian group G such that G=AB and |G|=|A||B|. Show that G=A^{-1}B. My attempt: If A is a subgroup, then A=A^{-1} and we're done. If A is not a ... 3 votes 2 answers 116 views ### If G=\langle H,K\rangle, then G'=[H,K] Let H,K be two Abelian subgroups of a finite group G such that G=\langle H,K\rangle. Show that G'=[H,K]. My attempt: Ofcourse, [H,K]\subseteq [G,G]=G'. Conversely, let x,y\in G. It ... 1 vote 0 answers 44 views ### Given a finite group G and N a normal subgroup such that N\cong K where K is Klein group and G/N\cong \Bbb{Z}/7\Bbb{Z}, prove N\leq Z(G) Given a finite group G and N a normal subgroup such that N\cong K where K is Klein group and G/N\cong \mathbb{Z}/7\mathbb{Z}, prove N\leq Z(G) I managed to deduce from both of the ... 1 vote 1 answer 22 views ### Group of isometries acting on a metric space is already discrete if a stabilizer is finite and an orbit is discrete My question is on page 163, the proof of Lemma 7 in the book Foundations of hyperbolic manifolds by John G. Ratcliffe. Let \Gamma be a group of isometries of a metric space X. If there is a point ... 2 votes 1 answer 48 views ### Prove G \cong H \times K when H,K \lhd G and the orders of H,K relatively prime with the product of their orders equaling the order of G Problem Statement Suppose G is a finite group. H,K \lhd G normal subgroups, \gcd(\lvert H \rvert, \lvert K \rvert)=1 and \lvert G \rvert = \lvert H\rvert \lvert K \rvert. Prove G \cong H \... 2 votes 1 answer 47 views ### Existence of a finite simple group that satisfies particular properties. Let G be a finite simple group and \tau_G = \{ o(x) : x \in G\}. Does there exist d_1, d_2 \in \tau_G that satisfy the following: d_1 < d_2 and d_1 does not divide d_2; for x, y \in ... 4 votes 1 answer 82 views ### Problem 5.17, Isaac's Character Theory Of Finite Groups I couldn't find how I should go to the result in the following problem. ( Problem 5.17, Isaac's Character Theory Book ) Let H \leq G and let \chi = (1_H)^{G}. Fix a positive integer n. For g \... 0 votes 1 answer 53 views ### Proving cyclic subgroup of a finite group is finite Prove that if G is finite then \forall a\in G, H:=\langle a\rangle is finite. Pf: Let G be a finite group with order m and let a \in G. Suppose that \langle a \rangle is infinite then ... 0 votes 0 answers 47 views ### magma gives an order 0 group? [closed] I want to create an elementary abelian subgroup C of order 16. But the following Magma code gives "#C = 0"... Where did it go wrong? F:=GF(3); A:=AlgebraicClosure(F); a:=RootOfUnity(8,A); ... 2 votes 0 answers 38 views ### Under which group conditions is the homomorphism g_2 f_1 :H_1 \to G\to H_2 an isomorphism? Let G be a group and H_1 ,H_2 be two r-images of G; i.e. there exist two homomorphisms f_i :H_i \to G and g_i :G\to H_i such that g_i f_i =id_{H_i}, for i=1,2. My question: Under which ... 0 votes 1 answer 28 views ### Let G_1,G_2 be groups, N a normal subgroup of G_1\times G_2. Suppose p_i(N)=G_i for all i\in\{1,2\}. Does it follow that N=G_1\times G_2? Let G_1,G_2 be groups and N a normal subgroup of G_1\times G_2. Let p_i\colon G_1\times G_2\to G_i be the projection with p_i(g_1,g_2)=g_i for i\in\{1,2\}. Suppose p_i(N)=G_i for all i\... -2 votes 0 answers 56 views ### Prove \operatorname{Dih}(n) \cong \langle a,b \mid a^2,b^n,abab\rangle. [duplicate] Prove \operatorname{Dih}(n) \cong \langle a,b \mid a^2, b^n, abab \rangle. My attempt: First, denote X={\{a,b\}}, \; R = \langle\!\langle a^2, b^n, abab \rangle\!\rangle, \; G = F_X/R. Let \phi:... 0 votes 0 answers 26 views ### Representation theory of Lie rings A Lie ring (M,+) is an abelian group with a product [\ ,\ ] (termed as the Lie bracket) satisfying [x,x]=0 [\ ,\ ] is bilinear [[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,y,z\in M. I want ... 3 votes 1 answer 54 views ### If G is a simple group, with \chi \in{\rm Irr}(G), such that \chi(1) = p, for some prime p, then G has a Sylow p-subgroup of order p. I am currently reading "Character Theory of Finite Groups". If the Sylow p-subgroup P is abelian, then by Theorem 3.13, p is the exact power of p diving |G: Z(G)| = |G| since G ... 3 votes 1 answer 55 views ### Given p odd prime and G=S_p and P\leq G p-Sylow and H=N_G(P) find |H| and prove if p=5 then H \cong C_4 \ltimes C_5  Given p odd prime and G=S_p and P\leq G p-Sylow and H=N_G(P) find |H| and prove if p=5 then H \cong C_4 \ltimes C_5 . For the first part we know that if we set n_p the number of ... 2 votes 0 answers 54 views ### Restrictions on the order of Finite Simple Groups I've just started learning about simple groups and I'm curious about restrictions on the order of finite simple groups. For example, I know that the only abelian finite simple groups are cyclic with ... 6 votes 0 answers 52 views ### Can the converse of Lagrange's Theorem hold for composite integers? [duplicate] In general the converse to Lagrange's Theorem is false - A_4 has no subgroup of order 6. However, the converse holds for the set of primes - given a prime p and a finite group G, if p ... 1 vote 1 answer 65 views ### Finding maximal, solvable, primitive subgroups of a large group in GAP I am trying to find the maximal, solvable, primitive subgroups of a large group N which is itself a subgroup of GL(n,p) for (n,p)=(4,3),(4,5),(4,7),(6,3),(10,3). However, GAP is too slow to run ... 2 votes 1 answer 73 views ### Prove a group of order 22,000 with 16 Sylow-5 subgroup has a normal (Sylow) 11 subgroup. Let G be a group of order 2^4\cdot 5^3 \cdot 11, H be a group of order 5^3 \cdot 11. Prove H has a normal 11-subgroup. Suppose n_5(G) < 16 (number of Sylow 5-subgroups of G), ... 0 votes 1 answer 38 views ### Let G be a finite group, |G|=n, there are x\in \mathbb{N} conjugacy classes. Prove there are xn homomorphisms from \mathbb{Z}^2 to G. Let G be a finite group, |G|=n, there are x\in \mathbb{N} conjugacy classes. Prove there are xn homomorphisms from \mathbb{Z}^2 to G. My approach is to find out the numbers of the normal ... -4 votes 0 answers 41 views ### Prove that \text{Hom}(G,C^*) is isomorphic to G if and only if G is abelian. Let G be a finitely generated group. Prove that \text{Hom}(G,C^*) is isomorphic to G if and only if G is abelian. My try: Suppose that \text{Hom}(G,C^*) is isomorphic to G. To show xy=yx ... 2 votes 1 answer 35 views ### GAP Program Efficiency for Number of Orbits Let S be a nonabelian finite simple group and p a prime divisor of |S|. I'm interested in finding the number of \textrm{Aut}(S)-orbits acting on the set \textrm{Cl}_{p'}(S), the set of all ... 0 votes 1 answer 63 views ### Orbits of a matrix group How to get the orbits of the action of the following matrix group on the standard basis of a 3-dim vector space? \begin{pmatrix} SL_2(2) & 0\\ * & 1\\ \end{pmatrix} where * denotes a 1\times... -2 votes 1 answer 86 views ### Let p be the smallest prime divisor of |G| and N\unlhd G  s.t. |N|=p. Find {\rm Aut}(N). [closed] Let p be the smallest prime divisor of |G| and N\unlhd G  s.t. |N|=p. Find {\rm Aut}(N). Attempt: We have |N|=p and since p is a prime number, N is a cyclic subgroup and so abelian, ... 4 votes 1 answer 90 views ### On Alperin's paper "The Green Correspondence and Brauer's Characterization of Characters" (aka what is a central factor?) I was studying the paper "The Green Correspondence and Brauer's Characterization of Characters" by J. Alperin and I couldn't understand two of the passages. Hypotheses and notations G is a ... 0 votes 1 answer 42 views ### Why D_4 is the biggest group generated by relations \langle f,r | f^2 =1 ,r^4 =1 , fr=r^3f \rangle? [duplicate] I want to find the presentation of group D_4= \{1, f, r,r^2,r^3, rf, r^2f, r^3f \} . r is the rotation of a square counterclockwise by 90 degree and f is the action that flips the square. Here ... 1 vote 0 answers 33 views ### Is any S_4-invariant function also S_6 invariant? Consider the following embedding of the permutation group S_4 inside S_6: \sigma \in S_4 \to \tilde \sigma \in S_6, where$$ \tilde \sigma\big(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}\big)=\big(...
I'm trying to prove that there are exactly 5 groups of order 42. My approach was to show that there is always a subgroup of order 6, let's say $H$, and a normal group of order 7, let's say $K$. It ...