Minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a group property (i.e., finitely generated group) satisfying a group metaproperty (i.e., extension-closed group property)
View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about finitely generated group |Get facts that use property satisfaction of finitely generated group | Get facts that use property satisfaction of finitely generated group|Get more facts about extension-closed group property

Statement

Quantitative version

Suppose G is a group, H is a normal subgroup, and G/H is the corresponding quotient group. If a denotes the minimum size of generating set for H and b denotes the minimum size of generating set for G/H, then the minimum size of generating set for G is at most a + b.

Corollary for finitely generated groups

Suppose G is a group, H is a normal subgroup, and G/H is the corresponding quotient group. Then, if both H and G/H are finitely generated groups, so is G.

Related facts