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Let G be a finite group. In general case, given two normal subgroups N and K of G, we need not to have N < K or K< N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume that G has such a property, that is the normal subgroups of G, constitute a chain with respect to inclusion. For example, simple groups, cyclic groups and symmetric groups satisfy this property. Certainly, if G has such a property, then the normal subgroups are necessarily characteristic. Also we may find out that the center of G must be cyclic. Indeed, every abelian group with this property must be a cyclic p-group (and vice versa). This also shows that G/G' is cyclic, for, the property in quotient hereditary. Please let me know, if these groups are studied before.

edited Mar 15 2011 at 12:49

asked Mar 14 2011 at 13:02

Amin
145●4

What is the question? – Emil Jeřábek Mar 14 2011 at 13:05

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Your statement "Indeed, every abelian group with this property is cyclic (and vice versa)" is not correct; only cyclic groups of prime power order have this property. – Tom De Medts Mar 14 2011 at 14:36

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A minor improvement: any nilpotent group with this property is cyclic. – ndkrempel Mar 14 2011 at 17:28

@ Tom Thank you Tom. You are right. I modifies it. – Amin Mar 15 2011 at 13:01

Jack Schmidt · Accepted Answer · 2011-03-14 21:38:56Z UTC

For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):

having a unique chief series,
every quotient group having a faithful primitive permutation action,
the upper Fitting series being a chief series
the lower Fitting series being a chief series

This is shown in:

Hawkes, Trevor O. "Two applications of twisted wreath products to finite soluble groups." Trans. Amer. Math. Soc. 214 (1975), 325–335. MR379657 DOI:10.2307/1997110

You might also be interested in the safari for zebra groups.

However, there are solvable groups with Frattini factors whose normal subgroups form a chain: SL(2,3) for example.

Groups in which normal subgroups consitiute a chain with respect to inclusion

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