Question

Let a,b be any two elements of a group. Prove that'

(ab)^{-1} = b^{-1}a^{-1}

Answer #1

let g be a group. let h be a subgroup of g. define
a~b. if ab^-1 is in h. prove ~ is an equivalence relation on g

2. Let a and b be elements of a group, G, whose identity element
is denoted by e. Prove that ab and ba have the same order. Show all
steps of proof.

(abstract alg) Let G be a cyclic group with more than two
elements:
a) Prove that G has at least two different generators.
b) If G is finite, prove that G has an even number of
generators

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.

1. Let a and b be elements of a group, G, whose identity element
is denoted by e. Assume that a has order 7 and that a^(3)*b =
b*a^(3). Prove that a*b = b*a. Show all steps of proof.

Let G be a group. Prove that following statements are
equivalent.
a.) G is commutative
b.) ∀ a,b ∈ G, (ab)2 = a2b2
c.) ∀ n ∈ N, ∀ a,b ∈ G, (ab)n =
anbn

(§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).

4. Let a, b, c be integers.
(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) =
1. (Hint: use the GCD characterization theorem.)
(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) =
1. (Hint: you can use the GCD characterization theorem again but
you may need to multiply equations.)
(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if
and...

Let O ∈ (AB) and C /∈ AB. Prove that there is a point D on the
same side of AB as C such that m(∠DOA) = m(∠COB).

2. Let G be a group containing 4 elements a, b, c, and d. Under
the group operation called
the multiplication, we know that ab = d and c2 = d. Which element
is b2? How about
bc? Justify your answer.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 8 minutes ago

asked 15 minutes ago

asked 21 minutes ago

asked 30 minutes ago

asked 30 minutes ago

asked 30 minutes ago

asked 39 minutes ago

asked 40 minutes ago

asked 40 minutes ago

asked 42 minutes ago

asked 43 minutes ago