By Makhnev A. A.

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Vn}. Zet H be the stabilizer of VI under the action of G . Then V1 is an F"H-module and V S V p . ) Proof. J W Because left multiplication by g , g E G , induces a permutation of the set {gi @ W , .. ,g, 8 W } ,and g(g1 8 W ) = g @ W , the assertion follows. (ii) Because G acts transitively on {Vl, V2,. . Vn} and H is the stabilizer of V1, it follows that n = ( G : H ) . ,g,, such that g;V1 = V,, 1 5 i 5 n. Since ) V? G 8 v1) CB ( 9 2 8 G ) CB ( g n 8 v1) and V=g,V1 @ * . $ g n V l as F-spaces, the map 0 : V F + V given by is a vector space isomorphism.

2, d e g x = (G : S ) for some subgroup S of G such that as is a coboundary. Thus (G : H ) I: degx for all irreducible a-characters x of G. Since p = CWHis a coboundary, FpH E F H so FPH has a one-dimensional mo'dule U. Setting W = U G , it follows that W is an F*G-module such that dimFW = (G : H ) 5 dimFX for any simple F"G-module X . But then W is a simple F*G-module which affords an irreducible a-character, say A , of minimal degree. Since degX = (G : H ) , the result follows. If n is a positive integer and p is a prime, then np denotes the highest power of p dividing n.

Then r dirnFUG = ( G : H ) = XdimFWi i=l (21 4 Degrees of irreducible projective characters 27 for some simple FaG-modules W1,... ,W,. Since dimFW; 2 degx for each i E (1,. . , r } , we deduce that d e g x 5 (G : H ) . Moreover, which shows that dJ(G: H ) . (ii) Since d l d e g x , the required assertion follows from (i). (iii) Assume that degx = (G : H ) . Then, by (2), degx = dimFUG and so U G is simple. Hence the required assertion follows by taking V = U G . Conversely, assume that degx = dimFV for some simple monomial FaGmodule V .

### 3-Characterizations of finite groups by Makhnev A. A.

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