Automorphism Groups Of Arithmetic P-Groups

In this paper, the arithmetic p-groups G_n,m of order p^n+m where p is odd primeand n > m≥1 are introduced from elementary number theory, which are not abelian.In the case where n≥2m, the automorphism group and central inner automorphismgroup are determined, the p-elements of automorphism group are described, and thenumber of fixed points in the group G_n,m of a p'-automorphism group is calculated.Theorem 1. Let G = pⁿ = b^pm = 1,a^b = a^1+pn-m> where p is an oddprime and n≥2m. Then Aut(G) = P×Q, where P is a normal Sylow p-subgroupof the automorphism group and |P| = p^n+2m-1, but Q is a cyclic group of orderp-1. Further the outer automorphism group is isomorphic to the semidirect productZ_p^m×U(Z_p^n-m, where the action of U(Z_p^n-m on Z_p^m is determined by the canonicalhomomorphism U(Z_p^n-m)→U(Z_Pm) (the definition of the canonical homomorphismdepends on the condition n- m≥m).Theorem 2. For anyσ∈Aut(G), let a^σ= b^jaⁱ,b^σ= ba^kpn-m, where (j, t, k)∈Z_p^m×U(Z_pⁿ×Z_p^m, thenσis a p-automorphism of G if and only if i is a p-elementin U(Z_pⁿ) and p^m-e|j, where p^e is the order of i modulo p^n-m.Theorem 3. The group of all inner central automorphisms of G is isomorphic to a subgroup of G.Theorem 4. The number of fixed points in G of any nontrivial p'-automorphism group of G is precisely p^m.For m = 1, G_n,1is exactly the finite non-commutative p-group with a cyclic maximal subgroup in [1], so the above theorems generalize the corresponding results in that paper.