Question

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!

Answer #1

Let G be a cyclic group, and let x1, x2 be two elements that
generate G . Show that f : G → G by the assignment f(x1) = x2 is an
isomorphism.

2.6.22. Let G be a cyclic group of order n. Let m ≤ n be a
positive integer. How many subgroups of order m does G have? Prove
your assertion.

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.

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 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?

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.

Let G = <a> be a cyclic group of order 12. Describe
explicitly all elements of Aut(G), the group of automorphisms of G.
Indicate how you know that these are elements of Aut(G) and that
these are the only elements of Aut(G).

(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)).

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