Question

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.

Answer #1

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

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.

2.6.22. Let G be a cyclic group of order n. Let m ≤ n be a
positive integer. How many subgroups of order m does G have? Prove
your assertion.

12.29 Let p be a prime. Show that a cyclic group of order p has
exactly p−1 automorphisms

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

For any prime number p use Lagrange's theorem to show that every
group of order p is cyclic (so it is isomorphic to Zp

Let p be a prime. Show that a group of order
pa has a normal subgroup of order
pb for every nonnegative integer b ≤
a.

Let n be an integer greater than 2. Prove that every subgroup of
Dn with odd order is cyclic.

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

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)).

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