TURING (ALAN)

Autograph letter signed ("A.M. Turing"), to Lionel March, discussing linear and group algebras, "The Computing Laboratory/ Manchester": 'DISTRIBUTIVE AND ASSOCIATIVE LAWS MUST HOLD' – ALAN TURING WRITES FROM 'THE COMPUTING LABORATORY, MANCHESTER', TO A YOUNG MATHEMATICAL PRODIGY
TURING (ALAN)
Autograph letter signed ("A.M. Turing"), to Lionel March, discussing linear and group algebras ("...A 'linear algebra' is a technical term for a space of vectors... To be a linear algebra as well as the addition of vectors and multiplication by scalars there must be multiplication of the algebra-elements together, and the distributive and associative laws must hold... What this all boils down to is that there are certain real coefficients... The distributive laws are then automatically satisfied, but for associativity it is necessary that [demonstrations follows]... In quite a lot of these systems each l/i l/j is another 'base element' e.g. l/k. In that case the base elements themselves form what is called a 'finite group', e.g. the six l/1 l/2 l/3 l/4 l/5 l/6 with multiplication table below would form a 'finite group'...When the base elements have this sort of multiplication table the algebra is called a 'group algebra'. Yours was a little different. Your base elements when multiplied together do not give another base element but +/- another base element. I was suggesting you should take the case where you allow any real factor, not just +/- 1. for instance the table [illustrated]... The problem I was suggesting you might try was to find all such 'group algebras with factors with six base elements..."); included in the lot is March's school mathematical notebook and his typed 'Rebmun' thesis (see below) 3 pages, slight dust-staining and weak at folds, 4to, "The Computing Laboratory/ Manchester" (2)