Gross provided answers to the questions of the exam the students had a few days before this lecture was held.
One question was as follows.
Q. Show that if a group has a unique element of order 2 then it is part of the center.
A. Order of element is equal to order of conjugate and because there is only one element of order 2 the following is true.
a = g a g^-1 for all g in G, or
a g = g a for all g in G,
and thus a belongs to the centre of G.
The lecture introduced subgroups of GL(n,F):
- the orthonogal group O(n,F) and
- the special linear group SL(n,F).
Where GL(n,F) consists of invertible matrices in O(n,F) this is further reduced to matrices with the property that the transposed matrix is equal to the inverse matrix. These matrices turn out to have determinants 1 or -1. ( Not true that all matrices with determinant 1 are orthogonal ). Matrices with +/- 1's on the diagonal are orthogonal as well as permutation matrices.
The elements of the Special Linear group are further reduced to those with a determinant value of 1.
A concrete orthogonal group is O(2,R) as subgroup of GL(2,R). This group consists of 2 by 2 matrices with elements from the real numbers. These matrices are linear transformations of vectors in R2.
More groups and geometry to follow.
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