pemutation groups - rigid motion of a cube
Find the order of the group of rigid motions of a cube.
permutation groups - rigid motion
A rigid motion of a cube can be thought of either as a permutation of its 8 vertices or as a permutation of its 6 sides. Find a rigid motion of a cube that has order 3, and express the permutation that represents it in both ways, as a permutation on 8 elements and as a permutation on 6 elements.
Show that in any group of permutations, the set of all even permutations forms a subgroup.
Let G1 and G2 be groups, and let G be a direct product of G1 x G2. Let H = {(x1, x2) element G1 x G2 | x2 = e} and let K = {(x1, x2) element G1 x G2| x1 = e}. (a) Show that H and K are subgroups of G. (b) Show that HK = KH = G (c) Show that H [see attachment] K = {(e,e)}.
Show that the null space of A^A coincide with the null space of A. What is the range?
Fields; Elements; Cyclic Groups
Find H K in {see attachment}, if H = <|3|> and K = <|5|>. This is all the problem says. I know the answer, but I do not know the reasoning.
Find H K in {see attachment}, if H={[1],[8]}, and K = {[1], [4], [10], 13], [16],[19]}. Thanks!
Inner / Linear Product; Polynomial Space
In a polynomial space of degree {see attachment} is (p,q)=p(0)q(0)+Ap(1/2)q(1/2)+Bp(1)q(1) for A,B>0. Define the linear product
Diagonizable Matrix; Inverse; Nullspace
1) If a Matrix A is diagonizable, must it have an inverse ? if so, is it diagonizable? Can {see attachment} be diagonized, does it have an inverse as well as {see attachment}
2) A is mxn
For m
Please see Attachment Respond in PDF format
