Show that every permutational product of a finite amalgam
am(A,B: H) is finite.Hence show that every...
Show that every permutational product of a finite amalgam
am(A,B: H) is finite.Hence show that every finite amalgam of two
groups is embeddable in a finite group.
Incorrect Theorem. Let H be a finite set of n horses. Suppose
that, for every subset S ⊂ H with |S| < n, the horses in S are
all the same color. Then every horse in H is the same color.
i) Prove the theorem assuming n ≥ 3.
ii) Why aren’t all horses the same color? That is, why doesn’t
your proof work for n = 2?
Using Kurosch's subgroup theorem for free proucts,prove that
every finite subgroup of the free product of finite groups is
isomorphic to a subgroup of some free factor.
Suppose {a1,...,am} is a complete set of representatives for
Z/mZ. Show:
(i) If (b,m)=1, then{b*a1,...,b*am}is a complete set of
representatives.
(ii) If (b,m)> 1, then{b*a1,...,b*am}is not a
complete set of representatives.
Let B be a finite commutative group without an element of order
2. Show the mapping of b to b2 is an automorphism of B. However, if
|B| = infinity, does it still need to be an automorphism?
Let B be a basis of Rn, and
suppose that Mv=λv for every v∈B.
a) Show that every vector in Rn is
an eigenvector for M.
b) Hence show that M is a diagonal
matrix with respect to any other basis C for Rn.
Let
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If b is
an element of G as well and the intersection of aH bH is non-empty
then aH and bH contain the same number of elements in G. Thus
conclude that the number of elements in H, o(H), divides the number
of elements...