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$\begingroup$

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?

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    It is a member of $C^4$ as you've defined it. What I mean is that you can take two elements ($a^3$ and $a^3$) and multiply them together to get something that, at first glance, does not appear to belong to the group ($a^6$).2012-09-23

2 Answers 2

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Try multiplying $a$ and $a^2$. Does it lie in your "subgroup"?

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    @Jim_CS That's okay. It's by asking questions that you will learn.2012-09-22
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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...