Volume : 6, Issue : 1, January - 2017

ABELIAN PROPERTIES OF SOLVABLE GROUPS AND ITS IRREDUCIBLE CHARACTER

Dr. Pankaj Kumar Chaudhary, Dr. Jawahar Lal Chaudhary

Abstract :

<p>&nbsp;<b style="text-align: justify;"><span style="font-size: 10pt; font-family: &quot;Times New Roman&quot;, serif;">Our main aim to obtain whether there exists any groups where e⁴ &minus;e³ &lt; |G| &lt; 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| &le; e⁴ &minus; 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 &minus; e³ &gt; |G|, then d &gt; e² &minus; e and all the nontrivial, normal subgroups of G are nonabelian.</span></b></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;text-align:&#10;justify;line-height:normal"><b><span style="font-size: 10pt; font-family: &quot;Times New Roman&quot;, serif;"><span style="letter-spacing:1.0pt"><o:p></o:p></span></span></b></p>