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Question #1

What is the definition of a Klein group? The [tex]K_4[/tex] group has a table that looks like this:

[tex]

\begin{array}{c|cccc}

*&e&a&b&c \\\hline

e&e&a&b&c\\

a&a&e&c&b\\

b&b&c&e&a\\

c&c&b&a&e

\end{array}

[/tex]

What is the strict definition of a Klein group? That every element generates the entire group? Is [tex]\langle \lbrace 1, 3, 5, 7 \rbrace, +\rangle[/tex] a Klein group?

Question #2

All subgroups of the cyclic group [tex]C_{24}[/tex] are cyclic groups [tex]C_n[/tex] where [tex]n \mid 24[/tex]. So to find all subgroups, one can locate all divisors for 24. Correct?

This is what I do not understand: The subgroups of [tex]C_24[/tex] with the order 24 is [tex]c, c^5, c^7, c^{11}, c^{13}, c^{17}, c^{19}, c^{23}[/tex]. What does [tex]c^n[/tex] mean? Does [tex]c^n[/tex] generate [tex]\frac{24}{\gcd(24, n)}[/tex] elements?

I would really, really appreciate some help with this!

Thanks in advance,

Nille

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# Klein and Cyclic group

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