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I mean, let $G$ be a group and has normal subgroups $N_1,N_2,N_3,...$

Let $N:max(N_1,N_2,N_3,...)$

Is the following true :

$$\bigcup_{k=1}^{\infty} N_k = N$$

I'm asking this because I'm concerned with the definition of Maximal normal subgroup.

A proper normal subgroup of $G$ is called maximal if I can't find a proper normal subgroup of $G$ that contains it? However this doesn't mean I can't find another normal subgroup which has order larger than.

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closed as unclear what you're asking by Batominovski, Chris Godsil, Arnaud D., user91500, Lord Shark the Unknown Apr 24 '17 at 6:41

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ What's $\max(N_1,N_2,N_3,\dots)$? $\endgroup$ – RJM Apr 23 '17 at 9:04
  • $\begingroup$ You have confused maximum element with maximal element. A maximum element is unique but a maximal element is not. Hence the notation $\max\{N1, N2, \dots\}$ is not justified. $\endgroup$ – Alex Vong Apr 23 '17 at 9:27
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In a general partially ordered set $(S,\leq)$, an element is called $x$ maximal if there does not exist $y$ such that $x < y$. For instance, if $S = \{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9\}$ and the order is the divisor order ($x_i \leq x_j$ if and only if $i$ divides $j$), then the elements $x_5, x_6, x_7, x_8, x_9$ are all maximal. This is much weaker than saying that for every $y$ we have $y\leq x$. Such an $x$ might be called maximum or greatest. In finite sets this is the same as saying $x$ is the unique maximal element. For example if we add the element $x_{5\cdot 7 \cdot 8 \cdot 9}$ to our set then this is the greatest element.

In group theory, the relation of inclusion $M\subseteq N$ is far more natural and useful than the weaker order relation $|M|\leq |N|$, and a subgroup is called maximal if it is maximal with respect to inclusion. As you say this is the same as not being contained in a larger subgroup.

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