SOME P-GROUPS WITH 2 GENERATORS WHICH SATISFY CERTAIN CONDITIONS ARISING FROM ARITHMETIC IN IMAGINARY QUADRATIC FIELDS

Let p be an odd prime. We give a list of certain types of p-groups G with two generators which satisfy the following two conditions (A) and (B): (A) [Ker V(G --> H): [G, G]] = [G:H] for the transfer homomorphism V(G --> H): G --> H/[H, H] of G to every normal subgroup H with cyclic quotient G/H, and (B) there exists an automorphism phi of G of order 2 such that g(phi+1) is-an-element-of [G, G] for every g is-an-element-of G. These conditions are necessary for G to be the Galois group of the second p-class field of an imaginary quadratic field. The list contains such a group that it may be useful for us to find an imaginary quadratic field with an interesting property on the capitulation problem.