Nilpotent groupsperiodic groupsLet G be a periodic group, and let LCM(G) be the set of all x is an element of G such that o(x(n)z) divides the least common multiple of o(x(n)) and o(z) for all z in G and all integers n. In this paper, we prove that the subgroup ...
Instances of this construction are found to occur naturally in the context of binary polyhedral groups. Recall that the core of a subgroup H of G is the intersection of the conjugates of H in G. For a permutation representation of a group G on a set Ω, let {H1,…,Hℓ} be a set...
Given a positive integer n, let np denote the highest power of p that divides n and let Cn denote a cyclic group of order n. By Clifford's theorem [4], a normal subgroup of a completely reducible group is also completely reducible. This fact will be used repeatedly. The following ...
A property of p-groups of nilpotency class p+1 related to a theorem of Schur In a p-group G of nilpotency class at most p +1, we prove that the exponent of the commutator subgroup gamma(2)(G) divides the exponent of G/Z(G). As a con... AE Antony,P Komma,VZ Thomas - 《...
istheindecomposableZ/p-moduleofdimensionn. 1.Introduction LetVbeafinite-dimensionalvectorspaceoverafieldk.Wechooseabasis, {x 1 ,...,x n },forthedual,V ∗ ,ofV.ConsiderafinitesubgroupGofGL(V).The actionofGonVinducesanactiononV
Further the translation complement modulo the subgroup of scalar coliineations is a dihedral group of order 24 and is the smallest when compared with all the planes reported so far. The translation complement of this plane divides the set of ideal points into 4 orbits of lengths 4, 4, 6 ...
Let R be a K-rational point on G of infinite order. Call n R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n R is the greatest positive integer which divides the order of ( R mod p ) for all but finitely many ...
The inertia group W I is now a reflection subgroup of W, generated by the reflections corresponding to the roots of (G,T) contained in the lattice ; in particular, W I ={1} if and only if x is a regular element of G. It follows (whether x is regular or not) that the height ...
The scaled consensus, which can be regarded as the generalized scenario of bipartite consensus, requires that the agents reach assigned proportions rather than a common value [18,19,20]. Cluster (group) consensus, which divides the corresponding agent set into separate groups, drives all agents ...
The scaled consensus, which can be regarded as the generalized scenario of bipartite consensus, requires that the agents reach assigned proportions rather than a common value [18,19,20]. Cluster (group) consensus, which divides the corresponding agent set into separate groups, drives all agents ...