Question

**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.**

Answer #1

Let G be a group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

Prove that, for any group G, G/Z(G) is isomorphic to Inn(G)

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 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 p,q be prime numbers, not necessarily distinct. If a group G
has order pq, prove that any proper subgroup (meaning a subgroup
not equal to G itself) must be cyclic. Hint: what are the possible
sizes of the subgroups?

True/False, explain:
1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=74
2. If |G|=19, then G is isomorphic to Z19.
3. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]
4. If F subset of K is a degree 5 field extension, viewing K as
a vector space over F, Aut(K, F) consists of...

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
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Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

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