Math Is Fun Forum

hayuning

Member

Registered: 2010-02-09

Posts: 1

abstract algebra (groups)

Someone please help me... I don't even understand.

1. Give an example of group G and two subgroup A, B of G such that AB is a subgroup of G

2. Prove that (1 2) cannot be written  as the product of disjoint 3 cycles

3. 1. if G has no proper subgrup, prove that G is cyclic

4. Express as the product of disjoint cycles and find the order

a. (1 2 3 5 7) (2 4 7 6)
b. (1 2) ( 1 3) ( 1 4)
c. (1 2 3 4 5) ( 1 2 3 4 6) (1 2 3 4 7)
d. ( 1 2 3) (1 3 2)

I would be very thankful if someone could help me..please...

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: abstract algebra (groups)

1. Hint: consider making A a subgroup of B.

2. Hint: Any 3-cycle is an even permutation.

3. If

and

, what is

?

4. I will show you my own example so you can do yours. Consider

. Work from right to left. Under this permutation:

So 1 maps to 3 and 3 maps back to 1, while 2 maps to 5 and 5 maps back to 2. Hence the permutation in disjoint cycles is

. You get the idea, hopefully.

The order of a product of disjoint cycles is the LCM of the lengths of the cycles.

Last edited by JaneFairfax (2010-02-10 00:05:09)