Question

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.

Answer #1

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.

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Let H be a normal subgroup of G. Assume the quotient group G/H
is abelian. Prove that, for any two elements x, y ∈ G, we have x^
(-1) y ^(-1)xy ∈ H

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Let a,b be any two elements of a group. Prove that'
(ab)-1 = b-1a-1

Let G be a simple graph with n(G) > 2. Prove that G is
2-connected iff for every set of 3 distinct vertices, a,
b and c, there is an a,c-path
that contains b.

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

(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

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)).

3. a) For any group G and any a∈G, prove that given any
k∈Z+, C(a) ⊆ C(ak).
(HINT: You are being asked to show that C(a) is a subset
of C(ak). You can prove this by proving that if x ∈ C(a), then x
must also be an element of C(ak) for any positive integer
k.)
b) Is it necessarily true that C(a) = C(ak) for any k ∈
Z+? Either prove or disprove this claim.

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