Answer to Question #118384 in Abstract Algebra for kila

Question

list all groups of order 2, and
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

Every group G has exactly one element identity e. Now, G is group of order 2, so there are two element in the group. Hence, there exist one element different from identity (say a).

We known that every element has unique inverse. Identity is of self inverse, so a is of self-inverse element $a = a^{-1} .$

So, There exist only one possibility of $G = \{e,a\}$ is group of order 2 where each element has self-inverse.

a) This group is cyclic because $a = a^{-1} \implies a^2 = e$ $\implies \exist a\in G : G = <a>$.

b) Since group G is cyclic so it is abelian also.

c) Subgroups of G are only {e} and G. Both are improper subgroup and hence normal subgroup of G.

d) Every group of order 2 is isomorphic to G. There is only one possibility of group.