Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 10 / b

Table of contents
- Solution b

Solution

The homomorphism law and

g g^{- 1} = g^{- 1} g = e

give

D (g) D (g^{- 1}) = D (e) = I, D (g^{- 1}) D (g) = I .

Thus

D (g^{- 1})

is both a left and right inverse of

D (g)

, proving

D (g^{- 1}) = [D (g)]^{- 1} .

If only multiplicativity is initially stated,

D (e)^{2} = D (e)

together with invertibility of representation matrices first implies

D (e) = I

. This explains why a group representation takes values in the general linear group rather than arbitrary square matrices.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version