The Generations And The Orders Of Automorphism Groups Of Some Groups Of Order P~6

The contents of this thesis mainly include the following two parts:One part mainly studies the generalizations and the orders of automorphism groups of some groups of orderp6. The concrete method is as follows:first, the thesis generalize the parameters of a given group of order p6, then research the properties of the group generated by these generators and defining relations, and verify the existence of the constructed group by using expansion theory and free group theory, mainly find the suitable expansion function and the appropriate parameters. If the group exists, the thesis prove its defining relations are consist-ent with our conjecture by using free goup method. Otherwise, the thesis constr-aint the conditions of the above generated formulas, then we determine the structure of the automorphism group of the above generated group by using the properties of finite group such as characteristic subgroup, normal subgroup, etc, and calculate the order of the corresponding autormorphism group by using congruence and combinatorial mathematical method. Finally, the thesis check whether the generated group satisfies the conditions of LA-group by the definiti-on or decision methods of LA-group, thereby get it under what conditions could be an LA-group. The other part solves the order of the automorphismgroup of a given group. Here, the thesis use a new method called matrix equation method, at the same time, we find a new very useful field and a vector space connected with it. This is a new and promising method. It not only provides a new thought to the research of the automorphism groups of finite groups, but also has certain applications in other related fields (such as computer technology, data mining, etc.).