ABELIAN PROPERTIES OF SOLVABLE GROUPS AND ITS IRREDUCIBLE CHARACTER

Dr. Pankaj Kumar Chaudhary; Dr. Jawahar Lal Chaudhary

Volume : VI, Issue : I, January - 2017

ABELIAN PROPERTIES OF SOLVABLE GROUPS AND ITS IRREDUCIBLE CHARACTER

Dr. Pankaj Kumar Chaudhary, Dr. Jawahar Lal Chaudhary

Abstract :

Our main aim to obtain whether there exists any groups where e⁴ −e³ < |G| < e⁴ +e³. Also, there is no concrete proof that the such group cannot exist. From assumptions of theorem, it is know that if such a group does exist, then all the normal subgroups of G must be nonabelian. we examine whether the bound |G| ≤ e⁴ − e³ can be proved when G is a simple group. Durfee and Jensen proved that if G has a nontrivial, abelian normal sub- group, then G has a normal subgroup N so that (G,N) is a p-Gagola pair for some prime p. Thus, if there exists a group G with e⁴ − e³ > |G|, then d > e² − e and all the nontrivial, normal subgroups of G are nonabelian.