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Part BCSIR NET December 2023union-of-proper-subgroups

Union of proper subgroups

Let G be any finite group. Which one of the following is necessarily true?

  1. A.G is a union of proper subgroups.
  2. B.G is a union of proper subgroups if |G| has at least two distinct prime divisors.
  3. C.If G is abelian, then G is a union of proper subgroups.
  4. D.G is a union of proper subgroups if and only if G is not cyclic.✓

Solution

A finite group is the union of its proper subgroups iff it is not cyclic: if it is cyclic a generator lies in no proper subgroup; if not cyclic every element generates a proper subgroup.

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: If d divides |G| then G has a subgroup of order d

From Groups › Subgroups, cosets, Lagrange, cyclic groups

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