On π-Subpairs of π-Blocks

On the number of B π′-characters in a nilpotent π-block

“…Based on π-block theory in [2][3][4][5][6][7][8][9], we will discuss the properties of nilpotent π-blocks as follows:…”

“…Based on the works of Slattery [2,3] , Robinson [4,5] and [6,7], the present paper generalizes the above idea to the π-block theory of a π-separable group, defines the nilpotency of a π-block, and proves that there is a unique B π -character in a nilpotent π-block.…”

“…In [6], we proved that G acts transitively on the set of maximal π-subpairs of a π-block B, that is G acts transitively on the set of Sylow B-subpairs of G. So G acts by conjugation on the set of π-subpairs of G, and the action keeps the order of inclusion of π-subpairs. Let (P, b P ) be a π-subpair of G, denote by N G (P, b P ) the invariant subgroup of (P, b P ) in G. Thinking of Lemma 1.1, we naturally define the nilpotency of a π-block as follows.…”

“…We say G is π-nilpotent if G is a semi product of a nilpotent Hall π-subgroup by a normal π -subgroup. The notations and conceptions are standard as in [1][2][3][4][5][6][7]. Lemma 1.1 [8] .…”

关于幂零<italic>π</italic>-块中<italic>B<sub>π'</sub></italic>-特征标个数的一个注记

“…Ð [1] ß, Broué Ä Puig ¥ p-Ø ¾¬, Ü Õ º p-Ø ÐÅ, ¬ ¥ p-Ø ÍÅÞ: ¥ p-Øß Á Brauer Ö©. Slattery [2,3] , Robinson [4,5] ĬÔÐ [6,7] ß Â ¬, ¤ » À π-×¼ π-Ø£ß, ¥ π-Ø AEܳ , ¬Ù¨: [1][2][3][4][5][6][7]. ¼ 1.1 [8] G Ð ¥ Hall π-¨ , Ò § 3 ℄:…”