Heng Lv, Wei Zhou and Dapeng Yu (see [LZY]) have been studying p-groups G all of whose cyclic subgroups are of index at most p2 in their normal closure in G. The main obstacle in these investigations has been two-generator subgroups {x, y} (x, y e G) such that 〈x〉 η 〈y〉 = {1} (see Lemma 144.4 and Lemma 144.5).

On the other hand, if G is a noncyclic p-group such that any two distinct maximal cyclic subgroups of G have intersection equal {1}, then we have either exp(G)= p or G = D2n, n > 3 (see Exercise 12 in Appendix 45). In fact, it is easy to prove this statement. Assume that exp(G) > p and let L G be a maximal cyclic subgroup of order > p. Let L M G with |M : L|= p. By Theorem 1.2, we get p = 2 and M is dihedral. Thus, all subgroups of G containing L and having order 2|L| are dihedral. Suppose that M G. Then N = NG(M) > M and N normalizes L. Let L0 be the cyclic subgroup of order 4 in L so that C = CN(L0) covers N/M, which gives C > L.Let K/L be a subgroup of order 2 in C/L. But then K is not dihedral, a contradiction. Hence we have M = G = D2n, n > 3, and we are done.

Therefore, it is of interest to classify p-groups G such that the following condition is satisfied:

In the first two Propositions 159.1 and 159.2 are determined p-groups G satisfying the condition (*), where G is either minimal nonabelian or metacyclic and p is any prime. It is clear that the condition (* ) says nothing in the case of p-groups of exponent p. But also in the case of 2-groups of exponent 4, the condition (*) says very little, and we characterize such groups in Theorem 159.3.

We recall that a p-group G is p-central if either p > 2 and Ω1 ( G) Z( G) or p = 2 and Ω2 ( G) Z( G). Also, we recall that a 2-group G is quasidihedral if it has an abelian maximal subgroup A of exponent > 2 and an involution t e G - A, which inverts each element in A.

In what follows G will be a 2-group of exponent > 4 satisfying the condition (*) .In Theorem 159.4 we determine up to isomorphism all such groups G which do not possess a normal elementary abelian subgroup of order 8. In Theorem 159.5 we classify our groups G with the assumption that Ü1(G) Z(G). It turns out that these groups must be 2-central, i.e., we must also have 02(G) Z(G). In Proposition 159.6 it is shown that if 02(G) is abelian, then G is also 2-central. Modular 2-groups (i.e., 2-groups that are D8-free) are classified in Theorem 159.7. Then we consider 2-groups G which do not have D8 as a subgroup or equivalently in which Ü1(G) is elementary abelian but 01(G) Z(G), and we show that such groups are of exponent 4 (Theorem 159.8). Finally, we classify in Theorem 159.9 our groups G in which Ü1(G) is nonabelian (i.e., D8 is a subgroup in G). The sta...