Nuclear (group theory) -Wikipedia

One field of mathematics, groups in group theory nuclear (This, English: core ) Is a regular part of a specific specific type of a group. The most commonly used is partial groups Formal nuclear And the group p -nuclear There are two types.

definition [ edit ]

group G In response to G Partial group H of Formal nuclear ( normal core ) or Heart ( core ) ^[first] And H include G The biggest regular part of the (or the same thing, H This is all the fellowship). More in general, G Partial set S Related to H What is the nucleus? S By H All of the intersections, that is,

${displaystyle mathrm {Core} _{S}(H):=bigcap _{sin S}{s^{-1}Hs}}$

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Is. In this wide definition S = G The nucleus on is regular nuclei. Regular package H ^G = ⟨ g ⁻¹ Hg | g ∈ G From the contrast with⟩ to the regular nucleus H _G Sometimes it is expressed ^[2] 。 The regular nuclei is consistent with itself to any regular part.

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The concept of regular nuclear weapons is important in the context of a group of groups. The regular nuclei of the partial part of each point is actively transformed throughout the orbit. Therefore, when the effect is transformed, the regular nuclear nucleus of any partial group is equal to the nucleus of action.

Non -nuclear group ( core-free subgroup ) Is a group where the regular nuclei is self -evident. That is, an obvious part of the iris is generated as a partial part of the transition and faithful group of groups.

The solution of the hidden partial group problem in the Abel group can be generalized, and the regular nuclei in the part of the arbitrary group can be obtained.

In this section G Is a finite group, but in some respects, it is generalized as a local owner or a sub -finished group.

definition [ edit ]

Prime number p In response to the finite group p -nuclear ( p-core ) Is its biggest regular p -It is defined as a partial group. This is an arbitrary Shiro of the group p -The regular nuclear nucleus of the part of the part

${DisplayStyle Text style bigcap _ {pin operatorname {syl} _ {p} (g)} p}$

It is. G of p -The nucleus is often O _p ( G ), And is especially used in the definition of the fitting part of the finite group. Similarly, group G of p ‘-nuclear ( p ′-core) Is that rank p And it seems like each other G In the biggest regular part of O _{p ′} ( G Expressed by) ^[4] 。 In the context of a large-scale non-unraveled group, including the classification theory of the finite simple group, the 2′-nucleus is often simply simply nuclear Call O ( G ) Write ^[4] 。 At this time, the “nucleus of the group” and “the nucleus of the part of the group” means a different meaning, so it is a little confusing. Further p ′, p -nuclear O _{p ′, p} ( G ) Is a group of surplus

${displaystyle O_{p’,p}(G)/O_{p’}(G)=O_{p}(G/O_{p’}(G))}$

Defined as. In the case of a finite group p ′, p -The nucleus is the only regular p -In become a group of zero.

Group p -The nucleus, the only Even formal ( subnormal ) p -The part can be defined as a part of the part. p ‘-Nuclear is the only regular p ‘-It is a part of the part, p ′, p -The nucleus is the only regular p -It is a group of zero.

Group p ‘-Nuclear and p ′, p -The nuclear weapons Rising p -List (upper p -series) It is the first item of. Group of prime numbers {π _first , Pi ₂ , …, Pi _{n +1} Partial group for} O _{Pi first ,Pi 2 ,…,Pi n +1} ( G ) of

${Displaystyle o_ {pi _ {1}, pi _ {2}, dots, pi _ {n+1}} (g)/o_ {pi _ {1}, pi _ {2}, dots, pi _ {n }} (G) = O_ {pi _ {n+1}} (g/o_ {pi _ {1}, pi _ {2}, dots, pi _ {n}} (g))}$

If you define it, π _{2 i −1} = p ′ And π _{2 i} = p By taking it p -A column is formed. Downhill p -There are also columns. Finish group p -Wei zero ( p -nilpotent) That is, it is your own p ′, p -At equal to nuclear weapons. Also, the finite group p -Dylodia ( p-soluble ) That is, it is my own rise p -The match is matched with any of the columns. A group G Raising p -The length of the column, G of p -length ( p -length) I call it. Furthermore, prime number p Finished group to G but p -The binding ( p -constrained) That means

${displaystyle C_{G}(O_{p’,p}(G)/O_{p’}(G))subseteq O_{p’,p}(G)}$

Say that it will hold.

Any 冪 zero group p -The zero, any p -The zero group is p -It is a clutter. A group of all kinds p -Insted, any p -The group is p -The binding. Group p -The necessary and sufficient conditions to be 冪 zero are that regular p -component (normal p -complement) Just p ′-Having something that matches the nucleus.

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As in the case of regular nuclear weapons, it is important to act on the set. p -Tuts and p ‘-Nuclear is important in modular expression theory of studying the actions of a group of vector space. Finished group p -The nucleus is a sign p Nuclear of any body expression expression on any body (kernel) It matches all fellowship. On the other hand, the finite group p ‘-Nuclear is the Lord p -Connects all the fitting of the regular (complex) that belongs to the block. Also, the finite group p ′, p -The nucleus is a target p Any body Lord p -The associated with all assigned expressions in blocks. Finished group p ′, p -The nuclear weapons are in the number p It matches the centralized group of abel primary factor (all the main factors belong to the main block). p -The assembly expression on the original body). In the case of a finite group p -Aligious clusis belongs to the main block p The necessary and sufficient conditions to become an assigned expression on the body of the body are the group p ‘-Nuclear is included in the nucleus of expression.

Can solve the foundation [ edit ]

As a similar concept, Can solve the foundation ( solvable radical ) O _∞ ( G ) Is defined as the biggest regular part of the idiom. Group p ‘-There are some different types of nuclear definitions that use some different types. For example, because of the good similarity with the 2’-nucleus, a group of non-possible is a group. G of p ‘-Nuclear, the root-rooted group p ‘-There are some documents defined as nuclear (Thompson) N -The papers about the group are examples, but thompson does not adopt this definition in subsequent research).

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