In: Advanced Math

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.

show that if H is a p sylow subgroup of a finite group G then
for an arbitrary x in G x^-1 H x is also a p sylow subgroup of
G

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?

Show that if a and b are integers with a ≡ b (mod p) for every
prime p, then it must be that a = b

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.

Hello! I am stuck on the following question: Show that every
simple subgroup of S_4 is abelian.

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...

Prove that every nontrivial finite group has a composition
series

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