Serial subgroup
From Encyclopedia of Mathematics
Let be a subgroup of a group
. A series of subgroups between
and
, or, more briefly, a series between
and
, is a set of subgroups of
,
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where is a linearly ordered set, such that
i) ,
for all
;
ii) ;
iii) is a normal subgroup of
;
iv) is a subgroup of
if
.
It follows that for all ,
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and
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and for a finite series, indexed by , hence
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A subgroup is called serial if there is a series of subgroups between
and
. If
is finite, a subgroup
is serial if and only if it is a subnormal subgroup. A subgroup
is called an ascendant subgroup in
if there is an ascending series of subgroups between
and
, that is, a series whose index set
is well-ordered.
References
[a1] | D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1 , Springer (1972) pp. Chapt. 1 |
How to Cite This Entry:
Serial subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serial_subgroup&oldid=48677
Serial subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serial_subgroup&oldid=48677