- #1
ToNoAvail27
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Homework Statement
Let ##G## be a group of order ##n## where ##n## is an odd squarefree prime (that is, ##n=p_1p_2\cdots p_r## where ##p_i## is an odd prime that appears only once, each ##p_i## distinct). Let ##N## be normal in ##G##. If I have that ##|G/N|=p_j## for some prime in the prime factorization of ##n##, and ##|N|=\frac{n}{p_j}##, then are the following true?
##N \cap G/N = \{e\}##
and
##N \sqcup G/N = G##
Homework Equations
The Attempt at a Solution
For the first claim, if you take ##N \cap G/N## and obtain a non-trivial element, that element will generate ##G/N##. And since that element is also in ##N##, I believe that would mean ##G/N < N##, but by LaGrange's theorem, since their orders are relatively prime, ##G/N## cannot be a subgroup of ##N##. Thus, the intersection is trivial. I'm a bit stuck on getting started on the second condition.