# Socle not is fully invariant in class two p-group

From Groupprops

## Statement

Let be a prime number. There exists a p-group of class two whose Socle (?) is not a Fully invariant subgroup (?).

## Related facts

- Center not is fully invariant
- Characteristic not implies fully invariant in odd-order class two p-group
- Center not is fully invariant in class two p-group
- Characteristic not implies fully invariant in class three maximal class p-group

## Proof

Let be a prime. Let be any non-abelian group of order with center (if , choose to be dihedral group:D8. Otherwise there are two possibilities for : a group of prime-square exponent, and a group of prime exponent). In all these groups, there is an element of order outside .

Define where is the cyclic group of order with generator . The socle of is the subgroup .

Then is not fully invariant in : Consider the retraction with kernel and with image generated by the element . This is an endomorphism of , but it does not send to itself, since the element gets sent to , which is outside .