Powering-invariant normal subgroup: Difference between revisions
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| [[Weaker than::quotient-powering-invariant subgroup]] || the quotient group is powered over any prime that the whole group is powered over. || [[quotient-powering-invariant implies powering-invariant]] || [[powering-invariant and normal not implies quotient-powering-invariant]] || {{intermediate notions short|powering-invariant normal subgroup|quotient-powering-invariant subgroup}} | | [[Weaker than::quotient-powering-invariant subgroup]] || the quotient group is powered over any prime that the whole group is powered over. || [[quotient-powering-invariant implies powering-invariant]] || [[powering-invariant and normal not implies quotient-powering-invariant]] || {{intermediate notions short|powering-invariant normal subgroup|quotient-powering-invariant subgroup}} | ||
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| [[Weaker than::finite normal subgroup]] || finite and a [[normal subgroup]] || || || {{intermediate notions short|powering-invariant normal subgroup|finite normal subgroup}} | | [[Weaker than::finite normal subgroup]] || finite and a [[normal subgroup]] || || || {{intermediate notions short|powering-invariant normal subgroup|finite normal subgroup}} | ||
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| [[Weaker than::normal subgroup of finite index]] || normal subgroup of [[subgroup of finite index|finite index]] in the whole group. || || || {{intermediate notions short|powering-invariant normal subgroup|normal subgroup of finite index}} | | [[Weaker than::normal subgroup of finite index]] || normal subgroup of [[subgroup of finite index|finite index]] in the whole group. || || || {{intermediate notions short|powering-invariant normal subgroup|normal subgroup of finite index}} | ||
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| [[Weaker than::normal subgroup contained in the hypercenter]] || || || || {{intermediate notions short|powering-invariant normal subgroup|normal subgroup contained in the hypercenter}} | | [[Weaker than::normal subgroup contained in the hypercenter]] || || || || {{intermediate notions short|powering-invariant normal subgroup|normal subgroup contained in the hypercenter}} | ||
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| [[Weaker than::endomorphism kernel]] || normal subgroup that is the kernel of an [[endomorphism]] || ([[endomorphism kernel implies quotient-powering-invariant|via quotient-powering-invariant]]) ||any normal subgroup of a finite group that is not an endomorphism kernel works. || {{intermediate notions short|powering-invariant normal subgroup|endomorphism kernel}} | |||
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| [[Weaker than::complemented normal subgroup]] || normal subgroup with a (possibly non-normal) complement || [(via endomorphism kernel) || || {{intermediate notions short|powering-invariant normal subgroup|complemented normal subgroup}} | |||
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| [[Weaker than::direct factor]] || normal subgroup with normal complement || (via complemented normal) || (via complemented normal) || {{intermediate notions short|powering-invariant normal subgroup|direct factor}} | |||
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Revision as of 20:33, 16 February 2013
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a powering-invariant normal subgroup if it is both a powering-invariant subgroup and a normal subgroup of the whole group. Here, powering-invariant means that for any prime number such that is powered over , we have that is also powered over .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| rationally powered normal subgroup | normal subgroup that is powered over all primes. | |FULL LIST, MORE INFO | ||
| quotient-powering-invariant subgroup | the quotient group is powered over any prime that the whole group is powered over. | quotient-powering-invariant implies powering-invariant | powering-invariant and normal not implies quotient-powering-invariant | |FULL LIST, MORE INFO |
| finite normal subgroup | finite and a normal subgroup | |FULL LIST, MORE INFO | ||
| normal subgroup of finite index | normal subgroup of finite index in the whole group. | |FULL LIST, MORE INFO | ||
| normal subgroup contained in the hypercenter | |FULL LIST, MORE INFO | |||
| endomorphism kernel | normal subgroup that is the kernel of an endomorphism | (via quotient-powering-invariant) | any normal subgroup of a finite group that is not an endomorphism kernel works. | |FULL LIST, MORE INFO |
| complemented normal subgroup | normal subgroup with a (possibly non-normal) complement | [(via endomorphism kernel) | |FULL LIST, MORE INFO | |
| direct factor | normal subgroup with normal complement | (via complemented normal) | (via complemented normal) | |FULL LIST, MORE INFO |