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Subgroups subnormal=>Nontrivial center - OIMU 2009 Problem 7

Source:

May 23, 2010

group theoryabstract algebrasuperior algebrasuperior algebra unsolved

Problem Statement

Let

G

be a group such that every subgroup of

G

is subnormal. Suppose that there exists

N

normal subgroup of

G

such that

H

is nontrivial and

G/N

is cyclic. Prove that

Z(G)

is nontrivial. (

Z(G)

denotes the center of

G

).
Note: A subgroup

H

of

G

is subnormal if there exist subgroups

H_1,H_2,\ldots,H_m=G

of

G

such that

H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G

(

\lhd

denotes normal subgroup).

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