A nonembedding theorem for finite groups

Abstract

Let N be the class of nilpotent groups with the following properties: (1) The center of N, Z j (N) is of prime order. (2) There exists an abeliani characteristic subgroup A of N such that Z1 (N) CA CZ2(N) where Z2(N) is the second term in the upper central series of N. The main result shown is the following: If NEI, then N cannot be an invariant subgroup contained in the Frattini subgroup of a finite group. Hobby has shown in [2] that a nonabelian group whose center is cyclic can not be the Frattini subgroup of any p-group. Chao in [I] has shown that a nonabelian group whose center is of prime order cannot be embedded in the derived group of any nilpotent group. The result obtained here is of a similar nature. All groups considered here are finite. DEFINITION. Let X be the class of nilpotent groups N which have the following properties: (1) The center of N, Z1(N), is of prime order. (2) There exists an abelian characteristic subgroup A of N such that Z1(N) CA CZ2(N) where Z2(N) is the second term in the upper central series of N. If N is an arbitrary group and N' is the derived group of N, then N'CnZ2(N) is an abelian characteristic subgroup of N. If N is nilpotent and N'nZ2(N)CZ1(N), then N'CZ1(N) and N has nilpotent length 1 or 2. Hence if N has nilpotent length greater than 2 and Z1(N) is of prime order, then NE&. The main result shown here is the following THEOREM. If NEX, then N can not be an invariant subgroup contained in the Frattini subgroup of any group. Clearly if the center of N is of prime order, then each term in the upper central series of N is a p-group. Futhermore Z2(N)/Z1(N) is elementary abelian. If NEX and A is as in the definition, then A = A/Z1(N) is elementary abelian and is therefore the direct product of cyclic groups of order p, denoted by ICr, , k.Cp. In the Received by the editors September 22, 1969. A MS Subject Classifications. Primary 2040, 2025.