By R. Goebel, E. Walker

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**Linear groups, with an exposition of the Galois field theory**

This vintage within the box of summary algebra has been referred to as "the first systematic therapy of finite fields within the mathematical literature. " Divided into sections-"Introduction to the Galois box conception" and "Theory of Linear teams in a Galois Field"-and full of examples and theorems, Linear teams continues to be suitable greater than a century after its book.

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Then the product F* @ F ' is a third irreducible K-representation of Clifford's theory of representations subgroups yields G~H G~HF.. 14 F := (F* @ F') t G~H. This theory also yields a result complete system of irreducible how we can obtain just a K-representations of G~H (cf. 15 If G is a finite group and closed field, then K an algebraically F = (F* @ F') t G~H runs exactly through a complete system of pairwise inequivalent and irreducible K-representations F of G~H, if runs through a complete system of pairwise not conjugate (with respect to K-representations of H') but irreducible G*, and, while F* is fixed, F" (cf.

Al~) of S2~Sn, then we obtain the associ- ated representation by forming its tensor product with the alternating representation of $2~S n with respect to the normal subgroup considered. 24 The alternating representations to S2~An, $2~S n representations of $2~S n An , $2~S n , respectively, A2 A2 (Inlo), (Oln), with respect are the (Olln), respectively. 25 (i) The ordinary irreducible representations of which are associated with (al~) $2~S n with respect to An S2~An, $2~S n , $2~S n A2 (a'l~') are: = (al~) ~ (Inlo), (~la) (~'I~') (ii) , respectively, A2 = (al~) | (oln), : (~I~ (olln)- | The ordinary irreducible representation $2~S n is selfassociated with respect to $2~S n , $2~S n (al~) of S2~A n , An A2 , respectively, if and only if A2 G = a' A ~ = ~, = ~', a = ~', respectively.

The inertia factor is ( S 2 . ( m [ S s ~ S 2 ] ~ / (S* N $2~S n A2 ) ~ SsNS2, so that the above extension produces the two irreducible representations 2] (o;2), and S S #[2]#[1 2] | (a;12) ' of the inertia group S2~(e[Ss~S2 ])A2. We notice that (c;2) = [a]#[m] | [ 2 ] ' , (C~;12) = [a]#[c] @ [12] ' yields s s (#[2]#[I 2 ] | ([~]#[a])') = @ L ((c;2) + ~ S2~(~[Ss~S2]) (a;12)) J extended to S2~(~[Ss~S2])A 2. 25 . e. the ordinary Hence (Celik/Pahlings/Kerber irreducible [I]) table of each is rational by Benson and Curtis [I]).

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