Let B be a finite commutative group without an element of order
2. Show the mapping...
Let B be a finite commutative group without an element of order
2. Show the mapping of b to b2 is an automorphism of B. However, if
|B| = infinity, does it still need to be an automorphism?
Let a be an element of a finite group G. The order of a is the
least power k such that ak = e.
Find the orders of following elements in S5
a. (1 2 3 )
b. (1 3 2 4)
c. (2 3) (1 4)
d. (1 2) (3 5 4)
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 and the intersection of aH bH is non-empty
then aH and bH contain the same number of elements in G. Thus
conclude that the number of elements in H, o(H), divides the number
of elements...
Let (G,·) be a finite group, and let S be a set with the same
cardinality as G. Then there is a bijection μ:S→G . We can give a
group structure to S by defining a binary operation *on S, as
follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) =
g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}.
First prove that (S,*) is a group.
Then, what can you say about the bijection μ?
Let G be a group of order 42 = 2 * 3 * 7
(a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow
3-subgroup of G . What are the orders of P3 and P7?
(b) Prove that P7 is the unique Sylow 7-subgroup of G and that
P7 is normal.
(c) Prove that P3P7 is a subgroup of G
(d) Prove that P3P7 is a normal subgroup of G .
(e) Let P2...
Let G be a group and a be an element of G. Let φ:Z→G be a map
defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group
homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is
a subgroup ofG.