1. ## Subgroups

Hi,

Using the subgroup test, or otherwise determine which of the following subsets H are subgroups of the stated group G, giving ur reasons:

a) H= {e, (12), (23), (13)} G= S3 (S sub 3)

b) H= {1, x, x^2, ...x^7} G= {1, x, x^2, ..., x^8}, the cyclic group of order 9

c) H= {A | det(A) is in Z} G=GL3(R) (L3= L sub 3, Z=integers)

d) H= {[k] | k even} G= Z/13Z, with group operation being addition.

Could u pls get me started.
Thanks!

2. ## H is not subgroup

a) H is not subgroup of S(3)

Sinse (1 2)(2 3)=(1 2 3) which does not belong to H, so the closure low is not satisfied

3. Originally Posted by choo
Hi,

Using the subgroup test, or otherwise determine which of the following subsets H are subgroups of the stated group G, giving ur reasons:

a) H= {e, (12), (23), (13)} G= S3 (S sub 3)

b) H= {1, x, x^2, ...x^7} G= {1, x, x^2, ..., x^8}, the cyclic group of order 9

c) H= {A | det(A) is in Z} G=GL3(R) (L3= L sub 3, Z=integers)

d) H= {[k] | k even} G= Z/13Z, with group operation being addition.

Could u pls get me started.
Thanks!
b.) $\displaystyle x,x^7\in H \implies x^8 \in H$...

d.) If $\displaystyle H$ was a group, then every element has an additive inverse. But $\displaystyle 2$ for example has no additive inverse since $\displaystyle 11 \not\in H$.