Minimum size of generating set of subgroup may be strictly greater than minimum size of generating set of group

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Statement

It is possible to have a finite group G and a subgroup H such that the minimum size of generating set for H is strictly greater than the minimum size of generating set for G.

Related facts

Facts used

  1. Every finite group is a subgroup of a 2-generated group, i.e., every finite group is a subgroup of a finite group with a generating set of size two. This follows from Cayley's theorem and the fact the symmetric group on a finite set is 2-generated.
  2. There exist finite groups whose minimum size of generating set is strictly greater than 2. For instance, elementary abelian group:E8 has a minimum size of generating set equal to three, because it is a three-dimensional vector space over field:F2 and generating sets for it as a group are the same as generating sets for it as a vector space.

Proof

The proof follows directly by combining Facts (1) and (2).