Universal completions of the cyclic amalgams of the same type

Abstract

Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p$^n$ for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL$_m$ (e$_q$) Also Aut(G)=p$^{(n-1)}$ $^{(m2)}$ GL$_m$ (Z$_q$). If $\alpha$is an automorphism of S$_n$and t is a transposition of S$_n$for n $\not$ 6, then $\alpha$(t)is a transposition. If $\alpha$ maps transposition to a transposition, then $\alpha$ is an inner automorphism. Then AUT (S$_n)$ $\simeq$ S $_n$ $\not$ 6. Furthermore, there exists an outer automorphism of S$_6$and OUT (S$_6$) $\simeq$ $\frac{z}{zz}$. Thus OUT (S$_g$)=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S$_n$S$_n$S_(n-1), $\phi$$_1$, $\phi$$_2$) where $\phi$ is an identity map from S_(n-1) to (S$_n$for i=1, 2 and n$\not$ 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral