Question

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

Answer #1

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.

prove that if G is a cyclic group of order n, then for
all a in G, a^n=e.

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.

Prove that any group of order 9 is abelian.

LetG be a group (not necessarily an Abelian group) of order 425.
Prove that G must have an element of order 5.

a) Prove: If n is the square of some integer, then n /≡ 3 (mod
4). (/≡ means not congruent to)
b) Prove: No integer in the sequence 11, 111, 1111, 11111,
111111, . . . is the square of an integer.

Let G be a group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

is it cyclic? Consider the abelian group (Z,∗) under the
operation a ∗ b = a + b − 1 for all a, b ∈ Z.

Let a be an element of order n in a group and d = gcd(n,k) where
k is a positive integer.
a) Prove that <a^k> = <a^d>
b) Prove that |a^k| = n/d
c) Use the parts you proved above to find all the cyclic
subgroups and their orders when |a| = 100.

Let n be an integer. Prove that if n is a perfect square (see
below for the definition) then n + 2 is not a perfect square. (Use
contradiction) Definition : An integer n is a perfect square if
there is an integer b such that a = b 2 . Example of perfect
squares are : 1 = (1)2 , 4 = 22 , 9 = 32 , 16, · ·
Use Contradiction proof method

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