Abstract Algebra: Let H={(g,g)}, g in G a group. Show H is a

[Jamers328]

Let G be a group, and let H = {(g,g) | g is in G}. Show H is a subgroup of G (+) G (this subgroup is the diagonal of G (+) G). When G is the set of real numbers under addition, describe G (+) G and H geometrically.

(+) is a + with a circle around it... called a cross sometimes? We just call it "plus" in my class, hopefully you know what I mean.

There is a hint in my book that states: (g,g)(h,h)^-1 = (gh^-1, gh^-1) but I don't really understand it. Any help would be appreciated. Thank you.

 

[Unco]

Re: Abstract Algebra: subgroup

The hint is alluding to the fact that a nonempty subset H of a group G is a subgroup of G iff \(\displaystyle ab^{-1}\in H\) whenever a and b are in H. Even ignoring that and returning to the basic definitions of a group and a subgroup, what is the indentity of H? What is the inverse of an element (g,g)? Etc.

Show us your complete work if you get stuck.