Mathematics 566
Homework (due Sep. 26th)
A. Hulpke
D&F;3.1: 3,11,33,36,41
D&F;3.2: 5,8,12,16
Cosets, normal subgroups and factor groups in GAP
The command IsNormal(G, U ) tests whether U is normal in G.
If N C G the command NaturalHomomorphismByNormalSubgroup(G, N ) returns a
homomorphism from G with kernel N , the image of this homomorphism is thus isomorphic to
the factor group G/N .
RightCoset(U, g) creates a coset with representative g, RightCosets(G, U ) returns the
set of right cosets U \G. RightTransversal(G, U ) returns a set of coset representatives.
1) (GAP) Let G = S4 and V = h(1, 2)(3, 4), (1, 3)(2, 4)i. Show that V C G. Compute the
multiplication table for the cosets of V in G (compute the cosets, multiply representatives, and
check with in, in which of the cosets the result lies).
2) Give an example of a group G and a subgroup U such that the multiplication of representatives does not give a well-defined multiplication of the cosets. (you can use GAP for calculations,
if you want)