User majid arezoomand - MathOverflow

Odd-order groups with homocyclic sylow subgroups

2013-05-05T15:21:17Z

We say that $G$ is a homocyclic group, if it is direct product of isomorphic cyclic groups. Is there any classification of finite odd-order groups which all their Sylow subgroups are homocyclic?

signs of eigenvalues

2013-04-21T18:29:52Z

Let $\Gamma$ be a multiple edge free (di)graph (with or without loop). Let $A$ be its adjacency matrix. It is clear that if $\lambda^2$ is an eigenvalue of $A^2$, then $\lambda$ or $-\lambda$ is an eigenvalue of $A$. What can we say about sign of $\lambda$ in general? I mean that can we exactly determine sign of eigenvalues of $A$ from eigenvalues of $A^2$?

Answer by majid arezoomand for the symmetric group $S_{2^{r−1}}$

2013-04-22T06:27:49Z

Yes. There is a method to find $p$-Sylow subgroups of $S_n$, by wreath product. See page 18 of the great book "Permutation groups, Dixon and Mortimer".

Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup?

2013-02-28T15:18:12Z

Let $\Gamma=Cay(G,S)$ be a Cayley graph over a group $G$, $H$ be a proper subgroup of $G$ and $\Sigma=Cay(H,T)$ where $S$ and $T$ are inversed-closed subsets of $G$ and $H$ not containing idendity, respectively. Is there any relation between $Aut(\Gamma)$ and $Aut(\Sigma)$ in general? When $T=H\cap S$, what can we say?

simply primitive permutation groups of degree $2p^2$

2013-01-30T19:14:45Z

I know that the only simply primitive permutation groups of degree $2p$, where $p$ is an odd prime, are $A_5$ and $S_5$. I want to know that: Is there a complete list of simply primitive permutation groups of degree $2p^2$, where $p$ is an odd prime?

finite abelian p-groups with solvable automorphism group

2012-12-29T17:02:54Z

Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?

Answer by majid arezoomand for solvable groups

2013-01-06T07:21:36Z

I just want to mention that, if $M$ be an abelian maximal subgroup of finte group $G$ then $G$ is solvable and its drived length is at most 3.(Exercise 3.4.7 of Permutation Groups, Dixon and Mortimer)

automorphism group of orbital graphs

2013-01-01T09:50:54Z

Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the orbital digraph $Graph(\Delta)$ is a digraph with vertex set $\Omega$ and edge set $\Delta$. Clearly, $G$ is a subgroup of automorphism group of $Graph(\Delta)$. Is there any paper or book that determine for which groups $G$, there is an orbital $\Delta$ that $G$ is equal to the automorphism group of $Graph(\Delta)$? Clearly, the action of $S_5$ on 2-subets of ${1,\ldots,5}$ is an example.

Answer by majid arezoomand for Exercises for group theory for physics

2012-12-27T20:46:20Z

I think that the H.F. Jones book "Group Theory in physics" can be another good reference.

Answer by majid arezoomand for composition factor of a group which isomorphic to the alternating group of order 7

2012-12-27T19:59:14Z

If you just want some examples, one of the examples is the symmetric group $S_7$ of degree 7. Clearly, you can costruct infinite groups that one of the its composition factors is $A_7$. For example $S_7\times\Bbb Z_p^r$ where $p$ is a prime and $r\geq 0$.

Answer by majid arezoomand for Spectral properties of Cayley graphs

2012-12-27T10:12:48Z

Just to keep the reference close to the question, I think that the manuscript of Petteri Kaski, Eigenvectors and Spectra of Cayley graphs,Helsinki university of technology, Spring Term 2002, is a very good reference.

simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$

2012-12-25T16:49:25Z

Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q_8$ be a semiregular normal subgroup of $G$?

Answer by majid arezoomand for finite groups with trivial frattini subgroup

2012-12-09T09:46:45Z

It seems that there are some examples. Let $G=D_{10}\times S_2$. Then the frattini subgroup of $G$ is trivial and the maximal subgroups are (that can be check by GAP): $M_1=\langle(6,7),(1,2,3,4,5)\rangle$, $M_2=\langle (2,5)(3,4),(1,2,3,4,5)\rangle$, $M_3=\langle(2,5)(3,4)(6,7),(1,2,3,4,5)\rangle$, $M_4=\langle(6,7),(2,5)(3,4)\rangle$, $M_5=\langle(6,7),(1,4)(2,3)\rangle$, $M_6=\langle(6,7),(1,2)(3,5)\rangle$, $M_7=\langle(6,7),(1,5)(2,4)\rangle$, and $M_8=\langle(6,7),(1,3)(4,5)\rangle$. Now consider $H_1=\langle(6,7)\rangle$ and $H_2=\langle(1,2,3,4,5)\rangle$.

Answer by majid arezoomand for finite groups with trivial frattini subgroup

2012-12-08T21:16:48Z

The answer in general is "No". For example consider the dihedral group $D_{10}$ of order 10. Then the frattini subgroup of $D_{10}$ is trivial and has 6 maximal subgroups that one of them is of order 5 and the others are of order 2. Now clearly, the answer is "NO".

vertex-transitive graphs of order 10 with full automorphism group $A_5$ or $S_5$.

2012-11-25T18:34:28Z

By a well-known result we know that a simply primitive permutation group of degree $2p$ where $p$ is a prime is $A_5$ or $S_5$ acting on 2-subsets of ${1,\ldots,5}$. The group has rank 3 and the orbital graph corresponding to the subdegree 3 is the Petersen graph with full automorphism group $S_5$. Is the left non-diagonal orbital graph a well-known graph? Is its full automorphism graph $A_5$? Generally is the Petersen graph (ignoring its complement) the only graph with 10 vertices with $S_5$ as its automorphism group? How many vertex-transive graph are with full automorphism group $A_5$ as a simply primitive group of degree 10?

Answer by majid arezoomand for imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime

2012-11-21T22:28:30Z

Thanks for every one who trid to answer the question. I have an answer for my question. Let $B\in\Sigma$. Consider the action of $K$ on $B$. If this action is faithful, then $K=S_2$. On the other hand $S_2= K'< K$. So the action of $K$ on $B$ is unfaithful. So by Lemma 2.1 of "On the Normality of Cayley Graphs of order $pq$, Z.P. Lu and M.Y.Xu, Australian Journal of Combinatorics, 27(2003) 81-93", $\Gamma$ is a lexicographic product. Now by Theorem 2.2 of "Cayley digraphs and lexicographic product, P. Xing and W.Dianjun,Front. Math. China 2007, 2(3): 1-8" the result is clear.

imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime

2012-11-21T06:39:37Z

Let $\Gamma=Cay(G,S)$ be a connected Cayley (di)graph over a group of order twice a prime and $\Sigma$ be a complete system of 2-blocks for $Aut(\Gamma)$. Let $K$ be the kernel of the action of $Aut(\Gamma)$ on $\Sigma$ and $K'$ be the kernel of the action of $R(G)$ on $\Sigma$. I need to prove that when $K'$ is a proper subgroup of $K$, then there is a subgroup of $H\leq G$ of order 2 that $S-H$ is union of some double cosets of $H$.

Semiregular subgroups of automorphism group of cayley graphs

2012-11-09T18:28:30Z

Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What about special cases?

Answer by majid arezoomand for Characteristic polynomial of hypercube graph

2012-11-11T04:23:57Z

The hypercube graph is a Cayley graph. The Hamming graph $H(n,r)$ is the Cayley graph $Cay(\Bbb Z_r^n,S)$ where $S$ is the set of all elements of $\Bbb Z_r^n$ with exactly one nozero coordinate. In particular, the Hamming graph $H(n,2)$ is the familiar $n$-dimensional hypercube.

Since $\Bbb Z_r^n$ is abelian, $\sum_{s\in S}\chi(s)$ where $\chi$ is an irreducible representation of $\Bbb Z_r^n$ is an eigenvalue of $H(n,r)$. The eigenvectors of the adjacency matrix $A$ of $H(n,r)$ are the vectors ${u_x}$, $x\in\Bbb Z_r^n$, where its $y$th coordinate is $\omega_r^{-\sum_{i=1}^nx_iy_i}$, $y\in\Bbb Z_r^n$ and $\omega_r=e^{\frac{2\pi i}{r}}$. Let $\lambda_x$ be the corresponding eigenvalue of $u_x$. If we dnote by $\omega_H(x)$ the number of nonzero coordinates in $x$, we have $\lambda_x=(r-1)n-r\omega_H(x)$. Now it is enough to put $r=2$. For more details on the spectrum of Cayley graphs see "Spectra of Cayley graphs, L. Babai, Journal of Combinatorial Theory, Series B 27, (1979) 180-189.

polycirculant conjecture

2012-11-05T09:06:51Z

By the polycirculant conjecture, every vertex-transitive graph is a polycirculant graph (D. Marusic 1981 and D. Jordan 1988). There are two papers that claim to prove this conjecture: 1. A. Golubchik, "On the polycirculant conjecture", available on http://arxiv.org/abs/math.GM/0204209, April 2002. 2. E. Mwambene, "A proof of the polycirculant conjecture", available on http://arxiv.org/abs/math/0506617, Jun 2005. But I find some papers that proved the conjecture in special cases, after 2005. For example (a) Every vertex-transitive graph of valency four is a polycirculant (E. Dobson et.al 2007) (b) All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism (M. Giudici and J. Xu 2007). (c) Every connected distance-transitive graph admits a semiregular automorphism (K. Kuntar and P.Sparl 2010).

So I want to know that the polycirculat conjecture is proved or not?

Comment by majid arezoomand

2013-05-05T15:47:05Z

I don't know, but a calssification of odd order groups with all Sylow subgroups abelian is also good.

Comment by majid arezoomand

2013-04-21T18:46:21Z

We can suppose that the eigenspaces of eigenvalues of $A^2$ are available.

Comment by majid arezoomand

2013-04-21T18:44:09Z

Ok. But I want a general result. Not for speciall graphs. Indeed I am looking for any result or paper concerning this problem.

Comment by majid arezoomand

2013-03-01T05:14:38Z

Thanks a lot for your responsibility. I am interest to finite groups.

Comment by majid arezoomand

2013-03-01T05:09:03Z

Thanks so much.

Comment by majid arezoomand

2013-02-28T20:17:22Z

Dear Prof. Godsil, thanks for your answer. I guess that when $T=H\cap S$, then one can give a decomposition of $Aut(\Gamma)$ (such as semidirect product, wreath product,...) which one of its factors is $Aut(\Sigma)$. Is this guess true?

Comment by majid arezoomand

2013-02-28T15:26:19Z

For example, If $G=\Bbb Z_4=<a>$, $H=<a^2>$, $S=\{a,a^3}$ and $T=H\cap S=\emptyset$, then $Aut(\Gamma)=S_2\wr S_2$ and $Aut(\Sigma)=S_2$ where $\wr$ denotes the wreath product of groups and $S_2$ is the symmetric group of 2 letters.

Comment by majid arezoomand

2013-02-28T15:22:42Z

I want a relation according to decomposition of groups (direct product, ...)

Comment by majid arezoomand

2013-01-14T06:38:29Z

Thanks so much for your nice answer.

Comment by majid arezoomand

2013-01-06T15:22:40Z

Certainly, your answer is very nice and I added only a reference for the question. Thanks for your previous comments.

Comment by majid arezoomand

2013-01-06T07:51:23Z

Every primitive permutation group with an abelian point stabilizer is regular of prime degree or a Frobenius group. Also by the prevous exercise (mentioned above) every finite primitive permutation group has a regular elementary abelian normal $p$-subgroup which is the same Frobenius kernel.

Comment by majid arezoomand

2013-01-06T07:03:12Z

Can you see also Exercise 3.4.7 of Permutation Groups, Dixon and Mortimer. To prove the result, just consider the action of $G$ on the right cosets of its (abelian) maximal subgroup and use its previous exercise.

Comment by majid arezoomand

2013-01-06T06:57:54Z

First you must type your question correctly. Also read the notatio 1.6 in page 24 of the book!

Comment by majid arezoomand

2013-01-02T07:39:34Z

Thanks for the reference. The thesis of Jing Xu, On closure of finite permutation groups, The university of Western Australia, 2005, is another good reference.

Comment by majid arezoomand

2012-12-31T16:34:19Z

A very nice proof!