Question

Characterize those integers n such that any Abelian group of order n belongs to one of exactly four isomorphism classes.

Answer #1

Characterize those integers n such that any abelian group of
order n belongs to one of exactly two isomorphism classes.

For which integers n such that 3<= n <=11 is there only
one group of order n (up to isomorphism)?
(A) For no such integers n
(B) For 3, 5, 7, and 11 only
(C) For 3, 5, 7,9, and 11 only
(D) For 4, 6, 8, and 10 only
(E) For all such integers n

Prove that any group of order 9 is abelian.

Let n be a positive integer. Show that every abelian group of
order n is cyclic if and only if n is not divisible by the square
of any prime.

If n is a square-free integer, prove that an abelian group of
order n is cyclic.

Let G be a non-abelian group of order p^3 with p prime.
(a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that
|NG(a)| = p^2 .
(c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t
forget to count the classes of the elements of Z(G)).

Let G be a group of order p^3. Prove that either G is abelian or
its center has exactly p elements.

Let G be a ﬁnite Abelian group and let n be a positive divisor
of|G|. Show that G has a subgroup of order n.

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Problem 7.
(i) Consider the cyclic group C18 of order 18. Determine
all the composition series of C18 then
verify the Jordan-Holder theorem for C18 (i.e. verify
that all those composition series have the same
length and the factors (after rearrangements) are isomorphic).
(ii) Give a composition series of A4.
(iii) Determine all the positive integers n such that the group Sn
is solvable.

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