Let 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 generated by LCM(G) is a ...
If G has a normal subgroup N such that μ(G)<μ(G/N), then we say that N is a distinguished subgroup of G, and that G/N a distinguished quotient. The group G is exceptional if it has at least one distinguished quotient Q; and we say also that G is an exceptional extension of...
Throughout the paper, p will always denote a prime. 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....
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 - 《...
1.Introduction LetVbeafinite-dimensionalvectorspaceoverafieldk.Wechooseabasis, {x 1 ,...,x n },forthedual,V ∗ ,ofV.ConsiderafinitesubgroupGofGL(V).The actionofGonVinducesanactiononV ∗ whichextendstoanactionbyalgebra automorphismsonthesymmetricalgebraofV ...
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 ...
In this paper, the impulsive multiple-bipartite consensus problem is discussed for networked second-order multi-agent systems (MASs) over directed network topology with acyclic partition. The definition of the multiple-bipartite consensus is introduced i