The hyperfocal subalgebra of a block
The hyperfocal subalgebra of a block
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DOI：
10.1007/s002220000072
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发表时间：
2000-08
影响因子：
3.1
通讯作者：
L. Puig
中科院分区：
文献类型：
--
作者：
L. Puig
1.1. 根据gr<s:1> n第一定理（参见[6]第7章第4.2节）可知，在有限群G中，通常称为Sylow p子群T与G的所谓派生子群的焦点子群T∩[G， G]的交集T∩[G， G]是由T中元素的G共轭决定的，即当u运行在T上时，当x运行在满足ux∈T的G的元素集合上时，由元素[x， u]生成，更准确地说，由Alperin的融合定理可知，当Q在T的子群集合上运行时，此交集由子群[NG (Q)， Q]生成（参见[1]）；实际上，同样的定理证明，用G的下中心级数的最后一项L (G)表示，当Q运行在T的子群集合上，x运行在NG (Q)的p元素集合上时，交集T∩L (G)是由子群[x， Q]生成的。换句话说，用S表示由这些换向子生成的T的子群——我们称它为T的超焦子群——可以证明：存在一个唯一的G的正规子群H，使得
1.1. It is known from Grün’s First Theorem (cf. Th. 4.2, Chap. 7 in [6]) that, in a finite group G, the intersection T∩[G, G]–usually called the focal subgroup–of a Sylow p-subgroup T with the so-called derived subgroup of G is determined by the G-conjugation of elements in T–namely it is generated by the elements [x, u] when u runs on T and x on the set of elements of G fulfilling ux∈ T. More precisely, it follows from Alperin’s Fusion Theorem that this intersection is generated by the subgroups [NG (Q), Q] when Q runs on the set of subgroups of T (cf.[1]); as a matter of fact, the same theorem proves that, denoting by L (G) the last term of the lower central series of G, the intersection T∩ L (G) is generated by the subgroups [x, Q] when Q runs on the set of subgroups of T, and x on the set of p-elements of NG (Q). In other terms, denoting by S the subgroup of T generated by these commutators–let us call it the hyperfocal subgroup of T–it can be proved that: there is a unique normal subgroup H of G such that