Question

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.

Answer #1

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.

13.
Let a, b be elements of some group G with |a|=m and |b|=n.Show that
if gcd(m,n)=1 then <a> union <b>={e}.
18. Let G be a group that has at least two elements and has no
non-trivial subgroups. Show that G is cyclic of prime order.
20. Let A be some permutation in Sn. Show that A^2 is in
An.
Please give me steps in details, thanks a lot!

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.

(A) Show that if a2=e for all elements a in a group
G, then G must be abelian.
(B) Show that if G is a finite group of even order, then there
is an a∈G such that a is not the identity and a2=e.
(C) Find all the subgroups of Z3×Z3. Use
this information to show that Z3×Z3 is not
the same group as Z9.
(Abstract Algebra)

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,...

1(a) Suppose G is a group with p + 1 elements of order p , where
p is prime. Prove that G is not cyclic.
(b) Suppose G is a group with order p, where p is prime. Prove
that the order of every non-identity element in G is p.

a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If
there are elements a, b ∈ G such that ab ∈/ H, then prove that
either a ∈ H or b ∈ H. (b) List the left and right cosets of H =
{(1), (23)} in S3. Are they the same collection?

the question says:
prove that if a is an element of a group G,
then the order of a = order of its inverse.
my attempt:
Let order of a=n , so aⁿ=e , and so (a)ⁿ(a^-1)ⁿ=e=(a^-1)ⁿ , so
order of a divides order of a^-1
let order of a^-1 =m. so (a^-1)^m=e if and only if a^m =e , so
order of a^-1 divides order of a
so they are equal.
Q.E.D
is the proof correct?

Let G be a group. g be an element of G. if
<g^2>=<g^4> show that order of g is finite.

Give the definition of group. Show that the identity (or
neutral) element of G is
unique. Moreover, show that the inverse of an element g 2 G is
unique as well.

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