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Unformatted text preview: 120 (a) (b) (c) 2. BASIC THEORY OF GROUPS For A 2 GL.n; R/ and b 2 Rn , deﬁne the transformation TA;b W
Rn ! Rn by TA;b .x / D Ax C b. Show that the set of all such
transformations forms a group G .
Consider the set of matrices
where A 2 GL.n; R/ and b 2 Rn , and where the 0 denotes a 1by-n row of zeros. Show that this is a subgroup of GL.n C 1; R/,
and that it is isomorphic to the group described in part (a).
Show that the map TA;b 7! A is a homomorphism from G to
GL.n; R/, and that the kernel K of this homomorphism is isomorphic to Rn , considered as an abelian group under vector addition. 2.4.21. Let G be an abelian group. For any integer n > 0 show that the
map ' W a 7! an is a homomorphism from G into G . Characterize the
kernel of ' . Show that if n is relatively prime to the order of G , then ' is
an isomorphism; hence for each element g 2 G there is a unique a 2 G
such that g D an . 2.5. Cosets and Lagrange’s Theorem
Consider the subgroup H D fe; .1 2/g Â S3 . For each of the six elements
2 S3 you can compute the set H D f
W 2 H g. For example,
.23/H D f.23/; .132/g. Do the computation now, and check that you get
the following results:
eH D .1 2/H DH .2 3/H D .1 3 2/H D f.2 3/; .1 3 2/g
.1 3/H D .1 2 3/H D f.1 3/; .1 2 3/g:
As varies through S3 , only three different sets H are obtained, each
Deﬁnition 2.5.1. Let H be subgroup of a group G . A subset of the form
gH , where g 2 G , is called a left coset of H in G . A subset of the form
H g, where g 2 G , is called a right coset of H in G .
Example 2.5.2. S3 may be identiﬁed with a subgroup of S4 consisting of
permutations that leave 4 ﬁxed and permute f1; 2; 3g. For each of the 24
elements 2 S4 you can compute the set S3 . ...
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