# Representations of GL(2) and SL(2) over finite fields by Garrett P. By Garrett P.

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2. Remark. 1 are available - see [Cohen 1989]. The above is the shortest and most elegant. J. Collins, H. Zieschang The conjugacy problem for F = ( X I - ) is also solvable. 3. Definition and Proposition. A word W == Xfl ... X~n is called cyclically reduced if it is reduced and X~n Xfl is not a peak. Let F = (X I -). (a) Every element of F is conjugate to an element represented by a cyclically reduced word. (b) Two cyclically reduced words represent conjugate elements if and only if one is a cyclic permutation of the other.

Then every element 9 of G can be uniquely expressed in the form g = aCIC2'" Cn, n ;::: 0, where a E A , and CI,"" Cn come alternately from transversals for the right cosets of A in G I and G 2 (and are not in A so that the case n = 0 arises only when 9 E A). 5. Theorem. Let G = ( Go, tiel Alt = A2 ) and let transversals for the right cosets of Al and A2 in Go be chosen. Then every element g of G can be written uniquely in the form btclCI ... 4. The existence of such a representation of 9 is easy.

Collins, H. Zieschang (N3) in a product of the form WI W 2W 3 == P I Z I Q1 1 . P2Q"2 I . P3Z3Q:;1 where QI = P2 ,Q2 = P3 and IWI W2 1 = IWII, IW2 W3 = IW3 1· Now WI W 2 == PI Z I Q"2 I and W2 W3 == P2 Z3 Q:;1 and no decrease in length occurs as a result of applying a Nielsen transformation (T2). However the properties of -< ensure that 1 and the induction hypothesis applies. 10. Proposition. If the systems (WI,"" W m ) and (W{, ... e. m = n and Wi = WI, 1:::; i :::; n. 11. Corollary. A finitely generated subgroup of a free group is free.

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