Groups and Ultra-groups

Abstract

<span style="text-decoration: underline;"><span style="color: #0000ff; font-family: Times New Roman; font-size: medium;">In this paper, in addition to some elementary facts about the ultra-groups, which their structure based on the properties of the transversal of a subgroup of a group, we focus on the relation between a group and an ultra-group. It is verified that every group is an ultra-group, but the converse is not true generally. We present the conditions under which, for every normal subultra-group of an ultra-group over a group, there exists a normal subgroup of that certain group. Moreover, by proving this feature that a monomorphism in groups preserve the ultra-groups over groups, we show that, corresponding to any monomorphism in groups, there is an ultra-group homomorphism on the subgroups of those groups. Finally, we prove isomorphism theorems for ultra-groups. These theorems connect three notions subultra-group, normal subultra-group, quotient ultra-group, and directly, similar to the isomorphism theorems in group theory and module theory are proved.</span></span>