Automorphisms Of The Unit Groups Of Square Radical Zero, Cube Radical Zero And Power Four Radical Zero Finite Commutative Completely Primary Rings

ABSTRACT

The study of automorphisms of algebraic structures has contributed immensely to

many important findings in mathematics. For example, Galois characterized the general

degree five single variable polynomials f over Q, by showing that the roots of such polynomials

cannot be expressed in terms of radicals, through the automorphism groups of

the splitting field of f. On the other hand, the symmetries of any algebraic structure are

captured by their automorphism groups. The study of completely primary finite rings has

shown their fundamental importance in the structure theory of finite rings with identity.

Quite reasonable research has been done towards characterization of the unit groups, R

of certain classes of finite commutative completely primary rings. Much less known however,

is whether there is a complete description of R, up to isomorphism. The existing

literature is still scanty on the characterization of Aut(R), the automorphism groups of

the unit groups of these classes of rings. Therefore, in this thesis, we have characterized

the structures and orders of the automorphisms of the unit groups of three classes of

commutative completely primary finite rings, that is, Square radical zero, Cube radical

zero and power Four radical zero finite commutative completely primary rings. The unit

groups of the classes of rings studied are expressible as R = Zpr−1 × (1 + J) such that,

Zpr−1 and (1 + J) are of relatively prime orders, where (1 + J) is a normal subgroup of

R and J is the Jacobson radical of R. We have expressed the structures of Aut(R) as

direct products of (Zpr−1) and GLrk(1+J)(Fp). We have made use of the invertible matrix

approach, the properties of diagonal matrices and determinants to count the number of

automorphisms of (1+J). We have then adjoined the counted Aut(1+J) to '((Zpr−1)),

where ' is the Euler’s phi-function, in order to completely characterize the order Aut(R).

Moreover, we have made use of the First Isomorphism Theorem to establish the relationship

between | GLrk(1+J)(Fp) | and | SLrk(1+J)(Fp) |. We noticed that our automorphisms

yielded very unique structure and order formulae, distinct from the well known structures

and order formulae of the automorphisms of the cyclic groups Cn. The results obtained

in this thesis contribute significantly to the existing literature on the structure theory of

finite rings with identity, thereby providing a much needed, accessible modern treatment

and a complete characterization of these classes of rings up to isomorphism.