Determination of Certain t-Groups

Abstract

Transiso graph \(\varGamma _d(G)\) is a graph defined for a finite group G and a divisor d of the order of G. Subgroups of G are vertices of \(\varGamma _d(G)\) and two subgroups are said to be connected by an edge if they have a pair of isomorphic normalized right transversals. A group G is called a t-group if graph \(\varGamma _d(G)\) is a complete graph for each divisor d of |G|. Completeness of transiso graphs for some groups like abelian groups, p-groups of order up to \(p^5\) where p is an odd prime, dihedral groups and dicyclic groups etc. are already discussed in the literature. In the present research article, we have discussed the completeness of transiso graphs for 2-groups of the order \(2^5\) and concluded that there are only two non-abelian t-groups (precisely dihedral group and generalized quaternion group) of the order \(2^5\) .

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