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Groupprops β

Having subgroups of all orders dividing the group order is not quotient-closed

Statement

It is possible to have a finite group G and a normal subgroup H such that:

  • G satisfies the property that it has subgroups of every order dividing the order of G.
  • The quotient group G/H does not have this property, i.e., there exists a positive divisor of the order of G/H such that G/H has no subgroup of that order.

Related facts

Proof

We can take G to be the direct product of A4 and Z2 and H to be the direct factor subgroup cyclic group:Z2.