User majid arezoomand - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T07:28:22Z http://mathoverflow.net/feeds/user/27831 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129727/odd-order-groups-with-homocyclic-sylow-subgroups Odd-order groups with homocyclic sylow subgroups majid arezoomand 2013-05-05T15:21:17Z 2013-05-15T09:29:18Z <p>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?</p> http://mathoverflow.net/questions/128272/signs-of-eigenvalues signs of eigenvalues majid arezoomand 2013-04-21T18:29:52Z 2013-04-22T07:27:22Z <p>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 $A^2$?</p> http://mathoverflow.net/questions/128311/the-symmetric-group-s-2r1/128312#128312 Answer by majid arezoomand for the symmetric group $S_{2^{r−1}}$ majid arezoomand 2013-04-22T06:27:49Z 2013-04-22T06:27:49Z <p>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".</p> http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a Is there any relation between automorphism group of a Cayley graph over a group and over its subgroup? majid arezoomand 2013-02-28T15:18:12Z 2013-02-28T22:01:36Z <p>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?</p> http://mathoverflow.net/questions/120349/simply-primitive-permutation-groups-of-degree-2p2 simply primitive permutation groups of degree $2p^2$ majid arezoomand 2013-01-30T19:14:45Z 2013-01-30T19:14:45Z <p>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?</p> http://mathoverflow.net/questions/117536/finite-abelian-p-groups-with-solvable-automorphism-group finite abelian p-groups with solvable automorphism group majid arezoomand 2012-12-29T17:02:54Z 2013-01-13T18:43:06Z <p>Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?</p> http://mathoverflow.net/questions/112198/solvable-groups/118187#118187 Answer by majid arezoomand for solvable groups majid arezoomand 2013-01-06T07:21:36Z 2013-01-06T07:21:36Z <p>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)</p> http://mathoverflow.net/questions/117774/automorphism-group-of-orbital-graphs automorphism group of orbital graphs majid arezoomand 2013-01-01T09:50:54Z 2013-01-02T17:24:49Z <p>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.</p> http://mathoverflow.net/questions/86336/exercises-for-group-theory-for-physics/117356#117356 Answer by majid arezoomand for Exercises for group theory for physics majid arezoomand 2012-12-27T20:46:20Z 2012-12-27T20:46:20Z <p>I think that the H.F. Jones book "Group Theory in physics" can be another good reference.</p> http://mathoverflow.net/questions/114250/composition-factor-of-a-group-which-isomorphic-to-the-alternating-group-of-order/117351#117351 Answer by majid arezoomand for composition factor of a group which isomorphic to the alternating group of order 7 majid arezoomand 2012-12-27T19:59:14Z 2012-12-27T20:08:57Z <p>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 $r\geq 0$.</p> http://mathoverflow.net/questions/18212/spectral-properties-of-cayley-graphs/117313#117313 Answer by majid arezoomand for Spectral properties of Cayley graphs majid arezoomand 2012-12-27T10:12:48Z 2012-12-27T10:12:48Z <p>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.</p> http://mathoverflow.net/questions/117202/simple-graphs-of-degree-16-with-a-semiregular-normal-subgroup-isomorphic-to-the-q simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$ majid arezoomand 2012-12-25T16:49:25Z 2012-12-25T19:49:57Z <p>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 $G$?</p> http://mathoverflow.net/questions/115830/finite-groups-with-trivial-frattini-subgroup/115878#115878 Answer by majid arezoomand for finite groups with trivial frattini subgroup majid arezoomand 2012-12-09T09:46:45Z 2012-12-09T21:33:01Z <p>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 $H_2=\langle(1,2,3,4,5)\rangle$.</p> http://mathoverflow.net/questions/115830/finite-groups-with-trivial-frattini-subgroup/115846#115846 Answer by majid arezoomand for finite groups with trivial frattini subgroup majid arezoomand 2012-12-08T21:16:48Z 2012-12-08T21:16:48Z <p>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".</p> http://mathoverflow.net/questions/114436/vertex-transitive-graphs-of-order-10-with-full-automorphism-group-a-5-or-s-5 vertex-transitive graphs of order 10 with full automorphism group $A_5$ or $S_5$. majid arezoomand 2012-11-25T18:34:28Z 2012-11-25T20:17:00Z <p>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?</p> http://mathoverflow.net/questions/114027/imprimitive-2-blocks-in-connected-cayley-digraphs-of-order-twice-a-prime/114111#114111 Answer by majid arezoomand for imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime majid arezoomand 2012-11-21T22:28:30Z 2012-11-21T22:35:38Z <p>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 $S_2= K'&lt; K$. 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.</p> http://mathoverflow.net/questions/114027/imprimitive-2-blocks-in-connected-cayley-digraphs-of-order-twice-a-prime imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime majid arezoomand 2012-11-21T06:39:37Z 2012-11-21T22:35:38Z <p>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 $H$.</p> http://mathoverflow.net/questions/111923/semiregular-subgroups-of-automorphism-group-of-cayley-graphs Semiregular subgroups of automorphism group of cayley graphs majid arezoomand 2012-11-09T18:28:30Z 2012-11-19T12:27:07Z <p>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?</p> http://mathoverflow.net/questions/112026/characteristic-polynomial-of-hypercube-graph/112061#112061 Answer by majid arezoomand for Characteristic polynomial of hypercube graph majid arezoomand 2012-11-11T04:23:57Z 2012-11-11T04:23:57Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/111537/polycirculant-conjecture polycirculant conjecture majid arezoomand 2012-11-05T09:06:51Z 2012-11-05T10:43:17Z <p>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 <a href="http://arxiv.org/abs/math.GM/0204209" rel="nofollow">http://arxiv.org/abs/math.GM/0204209</a>, April 2002. 2. E. Mwambene, "A proof of the polycirculant conjecture", available on <a href="http://arxiv.org/abs/math/0506617" rel="nofollow">http://arxiv.org/abs/math/0506617</a>, 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).</p> <p>So I want to know that the polycirculat conjecture is proved or not?</p> http://mathoverflow.net/questions/129727/odd-order-groups-with-homocyclic-sylow-subgroups Comment by majid arezoomand majid arezoomand 2013-05-05T15:47:05Z 2013-05-05T15:47:05Z I don't know, but a calssification of odd order groups with all Sylow subgroups abelian is also good. http://mathoverflow.net/questions/128272/signs-of-eigenvalues Comment by majid arezoomand majid arezoomand 2013-04-21T18:46:21Z 2013-04-21T18:46:21Z We can suppose that the eigenspaces of eigenvalues of $A^2$ are available. http://mathoverflow.net/questions/128272/signs-of-eigenvalues Comment by majid arezoomand majid arezoomand 2013-04-21T18:44:09Z 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. http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a/123269#123269 Comment by majid arezoomand majid arezoomand 2013-03-01T05:14:38Z 2013-03-01T05:14:38Z Thanks a lot for your responsibility. I am interest to finite groups. http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a/123258#123258 Comment by majid arezoomand majid arezoomand 2013-03-01T05:09:03Z 2013-03-01T05:09:03Z Thanks so much. http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a/123258#123258 Comment by majid arezoomand majid arezoomand 2013-02-28T20:17:22Z 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? http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a Comment by majid arezoomand majid arezoomand 2013-02-28T15:26:19Z 2013-02-28T15:26:19Z For example, If $G=\Bbb Z_4=&lt;a&gt;$, $H=&lt;a^2&gt;$, $S=\{a,a^3}$ 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. http://mathoverflow.net/questions/123224/is-there-any-relation-between-automorphism-group-of-a-cayley-graph-over-a-group-a Comment by majid arezoomand majid arezoomand 2013-02-28T15:22:42Z 2013-02-28T15:22:42Z I want a relation according to decomposition of groups (direct product, ...) http://mathoverflow.net/questions/117536/finite-abelian-p-groups-with-solvable-automorphism-group/118828#118828 Comment by majid arezoomand majid arezoomand 2013-01-14T06:38:29Z 2013-01-14T06:38:29Z Thanks so much for your nice answer. http://mathoverflow.net/questions/112198/solvable-groups/112202#112202 Comment by majid arezoomand majid arezoomand 2013-01-06T15:22:40Z 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. http://mathoverflow.net/questions/112198/solvable-groups/112202#112202 Comment by majid arezoomand majid arezoomand 2013-01-06T07:51:23Z 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. http://mathoverflow.net/questions/112198/solvable-groups/112202#112202 Comment by majid arezoomand majid arezoomand 2013-01-06T07:03:12Z 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. http://mathoverflow.net/questions/118184/notation-of-wilsons-book-the-finite-simple-groups Comment by majid arezoomand majid arezoomand 2013-01-06T06:57:54Z 2013-01-06T06:57:54Z First you must type your question correctly. Also read the notatio 1.6 in page 24 of the book! http://mathoverflow.net/questions/117774/automorphism-group-of-orbital-graphs/117788#117788 Comment by majid arezoomand majid arezoomand 2013-01-02T07:39:34Z 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. http://mathoverflow.net/questions/117608/a-question-on-almost-simple-groups/117649#117649 Comment by majid arezoomand majid arezoomand 2012-12-31T16:34:19Z 2012-12-31T16:34:19Z A very nice proof!