Register to reply

Abstract Algebra Questions....

by jbarrera
Tags: abstract, algebra
Share this thread:
jbarrera
#1
Nov1-11, 08:58 PM
P: 3
I have two problems that I'm a little puzzled by, hopefully someone can shed some light.

1) Show that if H and K are subgroups of the group G, then H U K is closed under inverses.

2) Let G be a group, and let g ε G. Define the centralizer, Z(g) of g in G to be the subset
Z(g) = {x ε G | xg = gx}.
Prove that Z(g) is a subgroup of G.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For problem 2 this is what I have but I am not sure if it is correct.

Since eg = ge for g in G, we know Z(g) is not the empty set.

-Take a in Z(g) and b in Z(g), and take any g in G, then we have...
(ab)g = a(bg) = a(gb) = (ag)b = (ga)b = g(ab). Thus ab is in Z(g).

- Take a in Z(g) and g in G. Then we know....
ag = ga
(a^-1* a )g = (a^-1 * g) a (multiplying both sides by a inverse)
e * g = a^-1 * g*a
g * a^-1 = a^-1 * g * (a * a^-1) ( multiplying again by a invese)
g * a^-1 = a^-1 * g

Thus a^-1 is in Z(g), so Z(g) is a subgroup of G.
Phys.Org News Partner Science news on Phys.org
Wildfires and other burns play bigger role in climate change, professor finds
SR Labs research to expose BadUSB next week in Vegas
New study advances 'DNA revolution,' tells butterflies' evolutionary history
Deveno
#2
Nov2-11, 02:48 PM
Sci Advisor
P: 906
what you did on 2 is fine. you could have saved a little time by showing b-1 is in Z(g) whenever b is, and then showing ab-1 is in Z(g) when a and b are, but not much.

for 1) x in HUK means:

x is in H...or
x is in K..or both.

so start by assuming x is in H, what can you say about x-1?

next, if x is not in H, it must be in K, and use a similar agument.


Register to reply

Related Discussions
Abstract Algebra-Pre Exam questions Calculus & Beyond Homework 0
Abstract algebra 2 questions Calculus & Beyond Homework 15
Abstract Algebra Questions - Need help for exam Calculus & Beyond Homework 4
Abstract Algebra Questions... Help Please Calculus & Beyond Homework 19
Questions concerning Finite Fields (Basic Abstract Algebra) Calculus & Beyond Homework 8