# mapping homomorphism and subgroup

• Jan 27th 2009, 07:55 AM
knguyen2005
mapping homomorphism and subgroup
1) Describe explicitly all homomorphisms h: C_6 maps to Aut(C_12)

Firstly, I have to find what group Aut(C_12) is isomorphic to, and then find the order of that group. Is this the right track?

If it is, then I don't know how to do after this.

How do you find the homomorphism?

2) Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that H =~ Q8 ( H is isomorphic to the quanterion group of 8)

What should I do for this question?

Thanks alot to all your help.
• Jan 27th 2009, 09:36 AM
ThePerfectHacker
Quote:

Originally Posted by knguyen2005
1) Describe explicitly all homomorphisms h: C_6 maps to Aut(C_12)

Firstly, I have to find what group Aut(C_12) is isomorphic to, and then find the order of that group. Is this the right track?

If it is, then I don't know how to do after this.

How do you find the homomorphism?

Hint: $\displaystyle \text{Aut}(C_{12}) \simeq \mathbb{Z}_{12}^{\times} \simeq \mathbb{Z}_2\times \mathbb{Z}_2$.

Quote:

2) Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that H =~ Q8 ( H is isomorphic to the quanterion group of 8)
Find elements $\displaystyle a,b$ with $\displaystyle |a|=|b|=4$ and $\displaystyle a^2 = b^2$ and then $\displaystyle bab = a^{-1}$. Once you do that the subgroup generated by these elements will satifisy the conditions of being isomorphic to the quaternion group.