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Define $P=\{M\subset G: M\unlhd G, M\neq G\}$.

Let $\mathscr{C}$ be a chain of $P$.

I have shown that $\bigcup \mathscr{C}$ is a normal subgroup of $G$, but i don't know how to prove $\bigcup\mathscr{C}\neq G$. Moreover, i'm not sure whether this is true.

Is it true? If so, how do i prove it is not $G$?

John. p
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  • You should look at http://math.stackexchange.com/questions/419091/does-every-infinite-group-have-a-maximal-subgroup and see the answers. – Jérémy Blanc May 13 '14 at 17:16

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Hint: Consider a non-trivial simple group, such as $A_n$ (the alternating group on $n$ letters) for $n \geq 5$.

Simon Rose
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  • I dont get what you mean with your hint. Here it works perfectly, since there is only the identity, so the union is not $G$. – Jérémy Blanc May 13 '14 at 17:14