\(\mathbb{Z}_2\)

»cyclic group of order two«
prove the following are true:

as a group it must be closed
per binary operations

one could, on this group, impose
extra structure (of a ring)

(but then its only ideal is itself)

(trivial) proper normal subgroup:
identity (that is, $\{\mathbb{I}\}$)

one more subgroup, also normal:
the group itself (but also trivial)

this group has just two characters,
only one of which is faithful

all groups are associative;
this one much more so than others

finite, rational, nilpotent,
strongly ambivalent, simple —

symmetric?