## Symmetry of Polycentric Systems by G. Fieck

By G. Fieck

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The contents of this booklet were used in classes given by means of the writer. the 1st used to be a one-semester direction for seniors on the collage of British Columbia; it was once transparent that stable undergraduates have been completely able to dealing with trouble-free staff conception and its software to easy quantum chemical difficulties.

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First note that if W is an RG-lattice, then K @ R W is a KG-module via Generalities 666 Furthermore, upon identification of W with its image 1 8 W , we see that W is a full RG-lattice in Ir' @R W . 6. Let W1 and WZ be two RG-lattices. If Ir' @ R W1 2 K @ R Wz, then w1 and w2 have the same composition factors (up to order of occure nce) . Proof. Put V = K @ R W2. Then W1 and WZcan be regarded as two full RG-lattices in the KG-module V. 5. 7. Let W be an RG-lattice and let K @R W = X @ Y be a nontrivial decomposition of KG-modules.

P,}, where cri and pj are the conjugates of (Y and p, respectively. Let H and A' be the stabilizers in G of cr and P , respectively. Then (G : H ) = m, (G : K ) = n and since ( m , n )= 1, a standard argument yields (G : ( H n I < ) ) = mn. But H n Ii' = G a l ( E / F ( a , P ) ) hence , ( E : F(cu,P)) = J H n KI It follows that as required. 4. Let E / F and K / F be finite field extensions, where F is a finite field. Then E @ F K is a field if and only if ( E : F ) and ( K : F ) are coprime. Proof.

Let {v;Ji E I} and {wjljE J } be F-bases of V and W , respectively. Then { l @ ( v ; @ w j ) lEi I , j E J} and { ( l @ v ; ) @ ( l @ w j )El iI , j E J} are E-bases of ( V @IF W ) , and VE W E , respectively. II, : 1 8 (wi @ wj) H (1 @ vi) C3 ( 1 C3 wj)extends to an E-isomorphism of (V W ) Eonto VE@,qW E . II,(1 8 ( v@ w))= (1 @ v)@ ( 1 @ w) for all v E V , w E W . Since for all g E G, and 8 Representations of direct products Given finite groups G and H , we seek to obtain information on the representations of G x H via those of G and H .