Hw8 Problem 10

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Problem 10

(*) Suppose $H$ and $K$ are subgroups of a group $G$. If $|H| = 12$ and $|K| = 35$, find $|H \cap K|$. Generalize.

Solution

Since $H \cap K \leq H$ and $H \cap K \leq K$, $|H \cap K| \bigm| |H|$ and $|H \cap K| \bigm| |K|$, so then $|H \cap K|$ must divide $12$ and $35$. Then $|H \cap K| \bigm| gcd(12, 35) = 1$, so $|H \cap K| = 1$. In general, for $|H| = m$ and $|K| = n$, $|H \cap K| \bigm| gcd(m, n)$. If $gcd(m, n) = 1$, then $|H \cap K| = 1$ and $H \cap K = \{e\}$.