Answer to Question #118380 in Abstract Algebra for kila

Question

for groups of order 1,
a)determine whether the groups are cyclic or not
b)determine whether the groups are abelian or not
c)list down all the subgroups and normal subgroups of the groups
d)list down if the groups are isomorphic to other groups

Answer 1

A group of order one is the trivial group containing one element. The trivial group can be {0} under addition or {1} under multiplication.

a)the trivial group is a cyclic group as it can be generated by the identity element.

b)the trivial group is a an abelian group because all cyclic groups are abelian.

c) the trivial group does not have subgroup .

d) There is only one unique group of order 1 up to isomorphism which is the trivial group.