Difference between revisions of "There exist subgroups of arbitrarily large subnormal depth"
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(New page: ==Statement== Let <math>k</math> be a positive integer. Then, we can find a group <math>G</math> and a subgroup <math>H</math> such that <math>H</math> is a <math>k</math>-[[subnormal sub...) |
(→Related facts) |
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+ | {{arbitrarily large subnormal depth statement}} | ||
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==Statement== | ==Statement== | ||
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with each <math>H_i</math> normal in <math>H_{i+1}</math>, and there exists ''no'' series of length <math>k-1</math> with the property. | with each <math>H_i</math> normal in <math>H_{i+1}</math>, and there exists ''no'' series of length <math>k-1</math> with the property. | ||
+ | |||
+ | ==Related facts== | ||
+ | |||
+ | * [[Stronger than::Normality is not transitive]] | ||
+ | * [[Weaker than::Descendant not implies subnormal]], [[Weaker than::there exist subgroups of arbitrarily large descendant depth]] | ||
+ | * [[Weaker than::Ascendant not implies subnormal]], [[Weaker than::there exist subgroups of arbitrarily large ascendant depth]] | ||
+ | * [[Weaker than::Normal not implies left-transitively fixed-depth subnormal]] | ||
+ | * [[Weaker than::Normal not implies right-transitively fixed-depth subnormal]] | ||
==Proof== | ==Proof== |
Latest revision as of 22:41, 19 April 2009
This is a statement of the form: there exist subnormal subgroups of arbitrarily large subnormal depth satisfying certain conditions.
View more such statements
Statement
Let be a positive integer. Then, we can find a group
and a subgroup
such that
is a
-subnormal subgroup of
but is not a
-subnormal subgroup of
. In other words, the subnormal depth of
in
is precisely
. Equivalently, there exists a series of subgroups:
with each normal in
, and there exists no series of length
with the property.
Related facts
- Normality is not transitive
- Descendant not implies subnormal, there exist subgroups of arbitrarily large descendant depth
- Ascendant not implies subnormal, there exist subgroups of arbitrarily large ascendant depth
- Normal not implies left-transitively fixed-depth subnormal
- Normal not implies right-transitively fixed-depth subnormal
Proof
Example of the dihedral group
Further information: dihedral group
Let be the dihedral group of order
. Specifically, we have:
.
Let be the two-element subgroup generated by
:
.
-
is
-subnormal in
. Consider the series:
.
Each subgroup has index two in its predecessor, and is thus normal. The series has length , so
is
-subnormal in
.
-
is not
-subnormal in
: To see this, note that the above subnormal series is a subnormal series of minimum length, because, starting from the right, each subgroup is the normal closure of
in the group to its right.