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14) (4 points) Prove or disapprove if the following digraphs are isomorphic? 3
10) (4 points) Prove or disapprove the function f:R → R such f(x) = 3x - 2 is one-to-one?
3. Suppose A is a noetherian ring. Prove that A is reduced if and only if it is isomorphic to a subring of a finite product of integral domains.
3. Suppose A is a noetherian ring. Prove that A is reduced if and only if it is isomorphic to a subring of a finite product of integral domains.
Prove that every two groups of order 3 are isomorphic to each other
4. Prove that a vector space V over F is isomorphic to the vector space L(F,V) of all linear maps from F to V. Note: We are not assuming V is finite-dimensional.
5) Leth-{ơes,lo(4) 4) That is, H is the set of permutation in S4 that leave the element 4 in its place. (i) Prove that H is a subgroup of S4. (ii) Prove that S is isomorphic to H. Explicitly give an isomorphism f: S3 → H listing the 6 elements of S, and giving the permutation in H to which it is sent under f. (ii) 1S "Spot check" the homomorphism property by showing that
5) Leth-{ơes,lo(4) 4) That is,...
Show that every model of incidence geometry which consists of precisely 3 points is isomorphic to the 3-point plane. (8 points)
Problem 4 (10 points) On the set R2, we define the following operation (ait a , a2eh + bge-ai). (ai, aa) * (b1, b2) = (i) Prove that (R2,*) is a group. (ii) Is this group isomorphic to (R2,+)?
7. (12 points) Prove that At and De (the dihedral group on the regular hexagon) are not isomorphic as groups
2. (40 points) Solve the "instant insanity" game depicted Solve the "instant insanity" game depicted below by graph theoretic techniques based on your class project. 3. (20 points) Prove that the two graphs below are isomorphic. You need to provide a function that has a one-to-one correspondence between the vertices of the two graphs. 2 6 5 4. (20 points) Prove that the two graphs below are isomorphic. You need to provide a function that has a one-to-one correspondence between...
Any group of order 4 is isomorphic to either C4 = {(1), (1234), (13)(24), (1432)}, the cyclic group of order 4, or K4 = {(1), (12)(34),(13)(24),(14)(23)}, the Klein-4 group (you don't need to prove this). Does there exist an onto homomorphism from D, onto C4? Does there exist an onto homo morphism from De onto K ? Justify your answers by either explicitly giving such a homomorphism, or proving that such a homomorphism cannot exist.