Question

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

Answer #1

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

A)
Prove that a group G is abelian iff (ab)^2=a^2b^2 fir any two
ekemwnts a abd b in G.
B) Provide an example of a finite abelian group.
C) Provide an example of an infinite non-abelian group.

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 11 minutes ago

asked 11 minutes ago

asked 17 minutes ago

asked 21 minutes ago

asked 29 minutes ago

asked 41 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago