- #1
Combinatus
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- 1
Homework Statement
Create a multiplication table for the group of invertible elements in the ring [tex]Z_{10}[/tex]. Can you rename the elements and arrange them so that the multiplication table is transformed into a multiplication table for the group [tex]Z_n[/tex] for some n?
Homework Equations
The Attempt at a Solution
If [tex]p \in Z_m[/tex], [tex]p[/tex] has an inverse iff [tex]GCD(p,m)=1[/tex], so the invertible elements in [tex]Z_{10}[/tex] are 1, 3, 7 and 9, and we end up with
[tex]\begin{bmatrix}
1 & 3 & 7 & 9\\
3 & 9 & 1 & 7\\
7 & 1 & 9 & 3\\
9 & 7 & 3 & 1\\
\end{bmatrix}[/tex]
as the suspiciously matrix-looking multiplication table in [tex]Z_{10}[/tex].
I don't know what the second sentence of the problem implies though. After attempting proof by asking IRC, I received the reply "Z/10Z =~ Z/2Z x Z/5Z -> (Z/10Z)* =~ (Z/2Z)* x (Z/5Z)* =~ Z/4Z", but I haven't seen any similar notation before. Where do I begin on this, or perhaps, what should I read to get a better understanding of similar problems?
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