Question

(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

Answer #1

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).

Let G be a group of order 4. Prove that either G is cyclic or it
is isomorphic to the Klein 4-group V4 =
{1,(12)(34),(13)(24),(14)(23)}.

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!

(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)

(Abstract algebra) Let G be a group and let H and K be subgroups
of G so that H is not contained in K and K is not contained in H.
Prove that H ∪ K is not a subgroup of G.

Suppose that G is a cyclic group, with generator a. Prove that
if H is a
subgroup of G then H is cyclic.

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.

prove that if G is a cyclic group of order n, then for
all a in G, a^n=e.

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Let
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If B is
an element of G as well show that aH and bH contain the same number
of elements in G.

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