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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 n∈Nn \in \mathbb{N} such that gpng^{p^{n}} = 1. Which of the following statements is FALSE?

  1. A.If |G| =p6= p^{6}, then G has a subgroup of index p2p^{2}.
  2. B.If |G| =p6= p^{6}, then G has at least five normal subgroups.
  3. C.Center of G can be infinite.
  4. D.There exists G with |G| =p6= p^{6} such that G has exactly six normal subgroups.✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests property not inherited.

See pricing

50 are analysed free — try those first.

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

The chapter behind this: Group actions, the class equation and p-groups — free to read

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

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