# Subgroups

• Mar 19th 2010, 12:14 PM
choo
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!
• Mar 19th 2010, 12:33 PM
fuzzy topology
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
• Mar 19th 2010, 01:23 PM
chiph588@
Quote:

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 $\displaystyle 11 \not\in H$.