Central factor not implies direct factor

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., central factor) need not satisfy the second subgroup property (i.e., direct factor)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about central factor|Get more facts about direct factor
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property central factor but not direct factor|

Statement

A central factor of a group need not be a direct factor.

Related facts

Proof

Let be cyclic group of order four and be the unique subgroup of order two, comprising the squares (or the elements whose order divides two). Then:

  • is a normal subgroup of .
  • is not a direct factor of : In fact, it is the only proper nontrivial subgroup of .