OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

The prime graph of a finite group G, which is denoted by GK（G）, is a simple graph whose vertex set is comprised of the prime divisors of ｜G｜ and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let P1 〈 P2 〈 … 〈 Pk be all prime divisors of ｜G｜- Then the degree pattern of G is defined as D（G） = （degG（pl）, degG（p2）,..., degG（pk））, where degG（p） signifies the degree of the vertex p in GK（G）. A finite group H is said to be OD-characterizable if G ≌ H for every finite group G such that ｜G｜ = ｜H｜and D（G） = D（H）. The purpose of this article is threefold. First, it finds sharp upper and lower bounds on v（G）, the sum of degrees of all vertices in GK（G）, for any finite group G （Theorem 2.1）. Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs （Propositions 3.1-3.7）. Third, it proves the linear groups L4（q）, q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable （Theorem 4.2）.

Rough similarity degree and rough close degree in rough fuzzy sets and the applications

Based on rough similarity degree of rough sets and close degree of fuzzy sets, the definitions of rough similarity degree and rough close degree of rough fuzzy sets are given, which can be used to measure the similar degree between two rough fuzzy sets. The properties and theorems are listed. Using the two new measures, the method of clustering in the rough fuzzy system can be obtained. After clustering, the new fuzzy sample can be recognized by the principle of maximal similarity degree.

OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U_6(2)

Let G be a finite group and π（G） = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F（G） as follows. The vertex set of this graph is 7r（G）, and two distinct vertices p, q are joined by an edge if and only if pq ∈ πe（G）. In this case, we write p - q. For p ∈π（G）, put deg（p） ：= ｜{q ∈ π（G）｜p - q}｜, which is called the degree of p. We also define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, where pl 〈 p2 〈 -……〈 pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a l-fold OD-characterizable group is simply called an OD-characterizable group. Let L ：= U6（2）. In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S3 is 5-fold OD-characterizable.

OD-characterization of Almost Simple Groups Related to U_3(5)

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.