Upper central series not is strongly central

From Groupprops
Jump to: navigation, search

Statement

The Upper central series (?) of a Nilpotent group (?) need not be a Strongly central series (?).

Definitions used

Upper central series

Further information: Upper central series

Strongly central series

Further information: Strongly central series

Related facts

Facts used

  1. Upper central series may be tight with respect to nilpotence class: For any natural number c, we can construct a nilpotent group G such that if Z_k(G) denotes the k^{th} member of the upper central series, the part of the upper central series upto Z_k(G) is also the upper central series of Z_k(G). In particular, Z_k(G) has nilpotence class k.

Proof

For c \ge 3, consider a group G that fits the situation of fact (1). Then, Z_2(G) has class exactly equal to two.

Suppose now that the upper central series of G were strongly central. Then, when numbered from G downwards, Z_2(G) is the (c-1)^{th} member, so by the definition of strongly central, [Z_2(G),Z_2(G)] is in the (2c-2)^{th} member from G downwards, which is the trivial subgroup since 2c - 2 \ge c + 1 for c \ge 3. Thus, Z_2(G) is Abelian.

This is a contradiction, so the upper central series of G is not strongly central.