Abstract Algebra Answers

2.5. If G is a finite group, show that there exists a positive integer m such that a^m = e for all a ∈ G.

2.4. If G is a group of even order, prove that it has an element "a\\ne e" satisfying a² = e.

2.3. Let G be a nonempty set closed under an associative product, which in addition satisfies:

(a) There exists an e ∈ G such that ae = a for all a ∈ G.

(b) Given a ∈ G, there exists an element y(a) ∈ G such that ay(a) = e.

Prove that G must be a group under this product.

Assume that the equation xyz = 1 holds in a group G. Does

it follow that yzx = 1? That yxz = 1? Justify your answer.

2.1. Let S be any set. Prove that the law of multiplication defined

by ab = a is associative.

If A is a finite set having n elements, prove that A has exactly

2ⁿ distinct subsets

Prove that there are infinitely many primes

Let "p_1,p_2,\\dots,p_n" be distinct pisitive primes. Show that "(p_1p_2\\dots p_n)+1" is divisible by none of these primes.

Show that if a nd b are positive untegers, then ab=LCM(a,b)*GCD(a,b)

If there are integers a, b, s, and t such that, the sum at+bs=1, show that GCD(a,b)=1