# Questions tagged [simple-groups]

Use with the (group-theory) tag. A group is simple if it has no proper, non-trivial normal subgroups. Equivalently (for finitely generated groups), its only homomorphic images are itself and the trivial group. The classification of finite simple groups is one of the great results of modern mathematics.

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### A finite nonabelian group is not simple if any two of its elements who are conjugate to each other commute.

Problem: Let $G$ be a finite nonabelian group. Assume any two elements $x,y \in G$ conjugate to each other also commute, show that $G$ is not simple. I write down a proof of this problem based on its ...
I would like to prove that if a non-comutative group $G$ has a subgroup $H$ whose index is equal to $3$ or $4$, then $G$ is not simple. In order to do, I took a group action $$\phi : G \times G/H \ni (... 0 votes 0 answers 21 views ### Schur multiplier of almost simple group Let S be a finite simple group. We call a finite group G an almost simple group with respect to S if S\leq G\leq \mathrm{Aut}(S). For a finite simple group, \mathrm{M}(S), the Schur ... 0 votes 1 answer 30 views ### On nonsplit noncentral extension of finite simple groups Let G be a finite group and N be its minimal normal subgroup such that G/N is a finite simple group. It is well know that if G=G' and N=Z(G), then N=\Phi(G). My Question is: Is it true ... 13 votes 2 answers 969 views ### What is the intuition behind simple Lie groups? What is the intuition behind simple Lie groups? Background: Simple groups and their final-dimensional representations are one of the huge improtance topics at my university course. My questions: Why ... 2 votes 1 answer 36 views ### Is the Socle of an almost simple group a simple group? Let G be a finite primitive group of degree n, and let H be the socle of G. Then if H is isomorphic to a direct power T^m of a nonabelian simple group T then the following holds when m=... 1 vote 0 answers 42 views ### Let G be a non abelian simple group. Show that {\rm Aut}(G^n) is isomorphic to {\rm Aut}(G) \wr{\rm Sym}(n). I got this question but don't know how to answer it. Let G be a non abelian simple group. Show that {\rm Aut}(G^n) is isomorphic to {\rm Aut}(G) \wr{\rm Sym}(n). I already know that {\rm Aut}(G)... 2 votes 1 answer 83 views ### The compactness theorem for simple groups I would like to use the compactness theorem to prove the following: Claim. If H is a subgroup of countable index in a simple group G, then there are finitely many conjugates of H with trivial ... 7 votes 0 answers 99 views ### Proof there are no perfect groups of order 3024 How can I prove that there are no perfect groups of order 3024? My attempt is the following: Each non-trivial finite perfect group admits a non-abelian simple quotient. This holds because if the ... 0 votes 1 answer 68 views ### Minimal normal subgroups of Product of simple groups Known Result: Let$$G= S_1 \times S_2 \times\dots\times S_n,$$where each S_i are non-abelian simple groups. Then S_i's are the minimal normal subgroup of G. (Even S_i's are the only minimal ... 1 vote 0 answers 36 views ### No group of order 160 is simple [duplicate] I'm trying to prove that no group of order 160 is simple. The following is my approach. Let G be a group of order 160. Note 160 = 2^55. I can easily get that n_2 = 1 or 5 and n_5 = 1 or ... 1 vote 1 answer 102 views ### If S is a simple subnormal subgroup of G. Prove that if S is nonabelian then S^G is a direct product of simple groups isomorphic to S. Here is the question and my solution. I understood the answer discussed here. My question and the solution is slightly different. Which does not use that T is non abelian. Proof : CLAIM-1: T^G is ... 4 votes 2 answers 96 views ### Group of order 1320 is not simple Group of order 1320 = 2^3\cdot 3\cdot 5\cdot 11 is not simple. Proof. Suppose there is a simple group G of order 1320. Then the number of Sylow 11 subgroup n_{11} = 12. Let G act on {\rm ... 0 votes 0 answers 44 views ### Showing Factor Groups are Simple; Subgroups of Solvable Groups are Solvable I'm aware this post is related to many other posts on proving that subgroups of solvable groups are solvable. However, there's a certain claim that is necessary to show based on the definition of ... 0 votes 0 answers 78 views ### If n\geq 5 prove that A_n is the only nontrivial normal subgroup in S_n. But please don't use the fact A_n is simple if n\geq 6. I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein. The following problem is Problem 17 on p.81 in this book. Problem 17: If n\geq 5 prove that A_n is the only nontrivial ... 6 votes 1 answer 123 views ### Does every quasisimple finite group have a faithful complex irrep? Every simple finite group has a faithful complex irrep. Indeed any nontrivial irrep of a simple finite group is faithful. That leads me to ask: Does every quasisimple group have a faithful complex ... 2 votes 1 answer 126 views ### There is no simple group of order 36. I tried to do this as an exercise and wanted to ask if my proof is correct or if it is missing something. Thank you so much. Let G be a group such that \lvert G \rvert = 36 = 2^2 \cdot 3^2. Show ... 2 votes 2 answers 123 views ### Finding a generating set of the simple group of order 168 I'm looking at the symmetries of some geometric object which I think should be the simple group G of order 168. I have in my possession symmetries \alpha, \beta, and \gamma of orders 2, 3, ... 3 votes 1 answer 48 views ### Is GL_n(\mathbb{F}_2) generated by a Jordan cell of size n and its transpose? Let n\geq 3. Consider the group GL_n(\mathbb{F}_2), and let J_n=\pmatrix{1 & 1 & & \cr & 1 & 1 & & \cr & & 1 & 1 \cr & & & \ddots&\ddots} ... 3 votes 0 answers 60 views ### Which finite groups can be proved simple via this theorem? The paper An Elementary Proof of the Simplicity of the Mathieu Groups M_{11} and M_{23} by Robin J. Chapman proves the following theorem: Let G be a transitive subgroup of S_p, and suppose |G|... 1 vote 1 answer 88 views ### If a chief factor of a group G is not simple, then G is not supersolvable. Is this true? If not, what is needed to make it true? The Details: A normal subgroup N of a group G, written N\unlhd G, is a subgroup N\le G such that for all g\in G, we have$$gN=Ng.$$A chief series of a group G is a finite set of normal ... 4 votes 1 answer 79 views ### Is there a reason the prime factors of |M_{24}| are all one less than the factors of 24? Wikipedia says of the Mathieu group M_{24}, a 5-transitive permutation group acting on 24 points,$$ |M_{24}|= 2^{10}\cdot3^3\cdot5\cdot7\cdot11\cdot23. $$The prime factors 2,3,5,7,11,23 are ... 3 votes 1 answer 162 views ### Growth of finite simple groups Background. There is an important group-theoretic notion of growth rate, defined for finitely-generated groups G equipped fixed finite generating sets S. The growth rate is (the equivalence ... 1 vote 0 answers 49 views ### If |G| = n < 60, and n is composite, then G is not a simple group. Why? [duplicate] If |G| = n < 60, and n is composite, then G is not a simple group. I am not totally sure how to solve this. So far, I have tried thinking of every possible theorem I can think of. A simple ... 0 votes 0 answers 43 views ### Proving that the symplectic group Sp_{2n}(K) is simple if |K|=2 or 3 I tried to prove the simplicity of G=Sp_{2n}(K) (n\geqslant 2), where K is any field. If |K|\geq 4, then this is fairly easy to prove. To start, we use the following result that is also used ... 1 vote 2 answers 51 views ### Assume H,G are simple groups. Can we prove that either H\unlhd HG or H\unlhd GH? Assume H,G are simple groups. Can we prove that H is normal in HG or in GH? Context: I want to use this argument in order to prove the following statement: If H_i and G_j are simple such ... 2 votes 1 answer 73 views ### Proving that the Spinor-kernel of SO3(K) is simple EDIT: This whole proof fails, as not everything in SO_3(K) can be written as \sigma_1\sigma_2. Instead, we have SO_3(K)\cong SL_2(K), which is not simple if |K|=3. I tried to prove something, ... 1 vote 2 answers 94 views ### How is C_4 constructed from the simple factor groups of its decomposition series I often hear that a every finite group is build up from simple groups (specifically the factors in its decomposition series, which are unique by Jordan-Hölder). The cyclic group of order 4 has a ... 3 votes 0 answers 89 views ### No simple group of order 2025 There is no simple group of order 2025. If G is simple then by the Sylow theorem, there must be 81 Sylow 5-subgroups and 25 Sylow 3-subgroups. Also, I can see that if Sylow 5 subgroups ... 1 vote 1 answer 52 views ### If G is a simple group of order 168, n_3=28, then a Sylow 3-subgroup T of N_G (P_3) acts transitively on the set of Sylow 7-subgroups of G  I am looking at Abstract Algebra, 3rd ed., by Dummit and Foote, page 208. In classfying groups of order 168, we first assume that there is a simple group of order 168 and prove that (1) G  has ... 4 votes 1 answer 92 views ### 2-generated finite non-Abelian simple groups and the existence of Hamiltonian cycles in their Cayley graph Given that G = \langle a, b\rangle and that a is an involution, when is it the case that there exists c, d such that G = \langle c, d\rangle and cd is an involution? At present, I am ... 3 votes 1 answer 152 views ### Number of conjugacy classes of elements of order 7 in a group of order 168 Let G be a simple group of order 168 (Here, we don't assume we know there is a unique such group). Compute the number of conjugacy classes of elements of order 7 in G [Hint: Consider Sylow 3-... 3 votes 0 answers 33 views ### Let G be a simple group, |G|=n, p a prime such that p|n. If G has more than n/(p^2) conjugacy classes then the p-Sylow sub are abelian Let G be a simple group, |G|=n, p a prime such that p\mid n. Prove that if G has more than n/(p^2) conjugacy classes then the p-Sylow subgroups are abelian. I think I have to use Sylow's ... 0 votes 0 answers 45 views ### Show that A is a simple group. [duplicate] The question is as follows: Let S_{\mathbb{N}} be the group of all permutation of \mathbb{N}. Let F be the subset of S_{\mathbb{N}} consisting of all permutations that fix all but a finite ... 1 vote 0 answers 58 views ### A detail in the proof that a group of order 168 is simple I am doing an excercise from Gallian's Contemporary Abstract Algebra that wants me to prove that the group PSL(2,Z_7) is simple. I have reached the point n_2=7, n_3=7 or 28 and n_7=8. I ... 1 vote 0 answers 45 views ### Intersection of normal subgroups and their indices exercise Suppose G is a finite group, H\leq G, and H is a simple group such that [G:H]=2. Prove: H is the only normal subgroup of G or there is a normal subgroup K\leq G, |K|=2 such that G=H \... 1 vote 1 answer 108 views ### Simplicity of O(p,q) Let \operatorname{O}(p,q) be the indefinite orthogonal group of signature (p,q) (over \mathbb{R}) and \operatorname{O}_0(p,q) its identity component. After doing a bit of research, I am given ... 2 votes 0 answers 142 views ### Prove that a group of order 540 is not simple Prove that a group of order 540 is not simple In this problem I have done the following steps. Sylow's Theorem gives n_3=10 and n_5=36. Let P be a Sylow 3 subgroup and Q a Sylow 5 ... 0 votes 1 answer 59 views ### A question in the answer of the following question : Prove that a group of order 72 can't be a simple group I had the following question: A group of order 72 can't be a simple group. As it was asked and user Dietrich Burde gave links to already asked questions. So, I got to know about : How to prove "... 2 votes 1 answer 55 views ### Let H\leq G where H is max simple. Prove either G is simple or there exists a minimal normal subgroup N of G such that G/N is simple. [duplicate] Question: Let G be a finite group and H a maximal, simple subgroup of G. Prove that either G is simple or there exists a minimal normal subgroup N of G such that G/N is simple. Thoughts:... 0 votes 1 answer 99 views ### Prove that there doesn't exists a simple group of order 24 and order 48 . [duplicate] Edit : SOme people are linking my question to solvability of group of order 8p but I don't want to use the concept of solvability of groups. So, Please don't link it to that. I have been following ... 5 votes 1 answer 164 views ### Non-abelian simple groups of odd order less than 10000. I am trying to solve problem 6.2.16 from Dummit and Foote, namely Prove there are no non-abelian simple groups of odd order < 10000. I did something similar for order <100, where I showed ... 0 votes 0 answers 28 views ### Are there any nonabelian simple groups of order 60? [duplicate] Prove that there are no nonabelian simple groups of order < 60. I am studying abstract algebra from Artin and trying some questions and got struck on this. Let on the contrary there exists a ... 0 votes 0 answers 147 views ### Simple group of order 60 is isomorphic to A_5 Prove that any simple group G of order 60 is isomorphic to A_5. I am studying abstract algebra from Artin and studying these assignment from a masters level math course. It gives hint. Use the ... 0 votes 1 answer 312 views ### N is a simple normal subgroup of a group G and G/N has a composition series... A question in section Normal and Subnormal Series of Hungerford Algebra. If N is a simple normal subgroup of a group G and G/N has a composition series, then prove that G has a composition ... 2 votes 1 answer 120 views ### Isaacs Algebra : A Graduate Course definition for diagonal subgroup I actually have a number of related questions. In Problem 7.2, Isaacs defines a subgroup D of G= M \mathop{\dot{\times}} N to be diagonal if$$ D \cap M = 1 = D \cap N \text{ and } DM = G = DN.  ...
I was reviewing an abstract algebra book, and I get the comment that "the monster group" (the set of matrix of size $196833 \times 196833$) has no normal subgroups,I wanted to know if anyone ...