I am a newcomer to group theory. I am looking at $C_4$ which has the elements $\{1,a,a^2,a^3\}$
Its subgroups are -
order 4: $\{1,a,a^2,a^3\}$
order 2: $\{1,a^2\}$
order 1: $\{1\}$
Why isn't $\{1,a,a^2\}$ a subgroup?
I am a newcomer to group theory. I am looking at $C_4$ which has the elements $\{1,a,a^2,a^3\}$
Its subgroups are -
order 4: $\{1,a,a^2,a^3\}$
order 2: $\{1,a^2\}$
order 1: $\{1\}$
Why isn't $\{1,a,a^2\}$ a subgroup?
Try multiplying $a$ and $a^2$. Does it lie in your "subgroup"?
Well, because $\,a\cdot a^2=a^3\notin \{1,a,a^2\}\,$ , so it isn't closed under the group operation!
Also, Lagrange's theorem tells us that any subgroup's order must divide the group's order...