- #1
nille40
- 34
- 0
Hi everybody!
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
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