User:Michellepoliseno /Math735 MIDTERM

(1) Let G be a group. Consider the subset $ H = {(x,x)| x \in G} \subset GxG. \ $

Which of the following claims is true:

(1) H is a subgroup of G,

(FALSE) H is not a subgroup of G, since in order to be a subgroup H must be a subset of G, but H is not a subset of G.

H is, however, a subgroup of GxG.

Since G is a group, $ e \in G \ $ and thus by definition of H, $ (e,e) \in H \ $, and therefore H is not empty. Let $ (x,x) \ $ and $ (y,y) \in H \ $ for some $ x, y \in G \ $. Since we found that $ (e,e) \in H \ $ we can say $ (y,y)^{1} \in H \ $. Then $ (x,x) \ $ x $ (y,y)^{-1} = (x,x) \ $ x $ (y^{-1},y^{-1}) = (xy^{-1},xy^{-1}) \in H \ $. Therefore H is a subgroup of GxG.

(2) H is a cyclic subgroup of G,

(FALSE) Since H is not a subgroup of G, it cannot be a cyclic subgroup of G.

(3) H is always a normal subgroup of GxG,

(FALSE)

(4) H is in general not a normal subgroup of GxG, but there are examples of noncommutative groups G such that H is a normal subgroup of GxG,

(FALSE)

(5) H is a normal subgroup of GxG if and only if G is commutative?

(FALSE)

(2) True of False:

(1) Every nontrivial group G contains a nontrivial proper subgroup H<G,

(2) Every nontrivial group contains a nontrivial normal proper subgroup,

(3) Every group H can be embedded (i.e. mapped by an injective homomorphism) into some group G so that G is larger than the image of H and the image of H is normal in G.

(FALSE)

Let $ f: H \to G \ $ be an injective homomorphism.

(3) Let $ f:R \implies S \ $ be a homomorphism of unitary commutativite rings and let $ I \subset R \ $ be an ideal contained in the ker(f). Show $ \exists! \ $ ring homomorphism $ g:R/I \implies S \ $ for which $ g \circ \mathcal \pi = f \ $ where $ \pi : R \implies R/I \ $ is the 'canonical' homomorphism, i.e., $ \pi(a) = (\overline{a}) \ $ (Notice: the problem has the existence and the uniqueness parts.)

Existence: Let $ a,b \in R \ $. Note since $ \pi : R \implies R/I \ $ is a homomorphism, then $ \pi (ab) = \pi (a) \pi (b) = (\overline{a})(\overline{b}) \ $ for some $(\overline{a}) \ $ and $ (\overline{b}) \in R/I \ $. And $ f(ab) = f(a)f(b) \ $ for some $ f(a), f(b) \in S \ $.

Then, $ g(\overline{a} \overline{b}) = f(ab) \ $, by definition of f. And $ f(ab) = f(a)f(b) \ $, since f is a homomorphism. Then, by definition of f, $ f(a)f(b) = g(\overline{a})g(\overline{b}) \ $, where $ g(\overline{a}), g(\overline{b}) \in S \ $. Thus g is a homomorphism.

Uniqueness:

(4) In the list below, identify all mutually isomorphic pairs of rings:

A = $ \C [X]/(X^{2}) \ $

B = $ \C [X]/((X-1)^{2}) \ $

C = $ \C [X]/(X^{3}) \ $

D = $ \C [X]/(X^{2}+1) \ $

E = $ \C X \C \ $

(5) In the list of rings in Problem (4), identify all mutually isomorphic pairs of F-vector spaces. (Notice: here we mean the naturally existing F-Vector space structures on any ring containing an isomorphic copy of F as a subring.)

(6) Let F be a field of positive characteristic $ p>0 \ $. In one of the homework assignments it was shown that the map $ F \to F \ $, $ a \to a^{p} \ $, is a ring homomorphism. Show that: (1) If F is finite, then the mentioned map is an automorphism of F. (2) Does part (1) extend to the general case when F is an arbitrary (not necessarily finite) field of characteristic $ p>0 \ $?