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Part BCSIR NET June 2023normal-subgroups-of-p-groups

Normal subgroups of p groups

Let p be a prime number. Let G be a group such that for each g ∈ G there exists an such that = 1. Which of the following statements is FALSE?

  1. A.If |G| , then G has a subgroup of index .
  2. B.If |G| , then G has at least five normal subgroups.
  3. C.Center of G can be infinite.
  4. D.There exists G with |G| such that G has exactly six normal subgroups.✓

Solution

A finite p-group has a normal subgroup of every order pᵏ dividing |G|, so |G| gives at least seven normal subgroups (orders — 'exactly six' is impossible. (1), (2) follow from the same fact; (3) an infinite abelian p-group (e.g. the Prüfer group) has infinite centre.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: Every group of order p² is cyclic

More on this topic

From Groups › Group actions, class equation, p-groups

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