Question

Let Z be the integers.

(a) Let C_{1} = {(a, a) | a ∈ Z}. Prove that
C_{1} is a subgroup of Z × Z.

(b) Let n ≥ 2 be an integer, and let C_{n} = {(a, b) | a
≡ b( mod n)}. Prove that C_{n} is a subgroup of Z × Z.

(c) Prove that every proper subgroup of Z × Z that contains
C_{1} has the form C_{n} for some positive integer
n.

Answer #1

Suppose n and m are integers. Let H = {sm+tn|s ∈ Z and t ∈
Z}.
Prove that H is a cyclic subgroup of Z.
......................
Please help with clear steps that H is a cyclic subgroup of
Z

Let a, b be integers with not both 0. Prove that hcf(a, b) is
the smallest positive integer m of the form ra + sb where r and s
are integers.
Hint: Prove hcf(a, b) | m and then use the minimality condition to
prove that m | hcf(a, b).

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Prove this statement. Z[i] is the Gaussian integers:
Let α ∈ Z[i]. Then α is a unit if and only if N(α) = 1.

Let a, b, c, m be integers with m > 0. Prove the following:
(a) ”a ≡ 0 (mod 2) if and only if a is even” and ”a ≡ 1 (mod 2) if
and only if a is odd”. (b) a ≡ b (mod m) if and only if a − b ≡ 0
(mod m) (c) a ≡ b (mod m) if and only if (a mod m) = (b mod m).
Recall from Definition 8.10 that (a...

Let S = {x ∈ Z : −60 ≤ x ≤ 59}.
(a) Which integers are both in S and 6Z?
(b) Which integers in S have 1 as the remainder when divided by
6?
(c) Which integers in S are also in −1 + 6Z?
(d) Which integers satisfy n ≡ 3 mod 6?

Prove: Proposition 11.13. Congruence modulo n is an equivalence
relation on Z :
(1) For every a ∈ Z, a = a mod n.
(2) If a = b mod n then b = a mod n.
(3) If a = b mod n and b = c mod n, then a = c mod n

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

Let R be the relation on Z defined by:
For any a, b ∈ Z , aRb if and only if 4 | (a + 3b). (a) Prove that
R is an equivalence relation.
(b) Prove that for all integers a and b, aRb if and only if a ≡
b (mod 4)

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