Find all non-isomorphic abelian groups of order 48
Prove that every two groups of order 3 are isomorphic to each other
4. Let G and H be isomorphic groups. Assume that every element of G has a square root (that is, for all IEG, there exists y E G such that I = y). Prove that every element of H has a square root.
2. (a) Let p > 2 be prime. Describe all groups of order p. (b) Give two examples of non-isomorphic groups of order p?, explaining clearly why they are non-isomorphic. (c) For which pairs of primes p >q is there a unique group of order pg? (a) Classify all groups of order 4907 explaining clearly all steps of your argument.
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3 is not an isomorphism. 4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3...
How many non-isomorphic unital rings are there of order 4? Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
Use the Fundamental Theorem of Finitely Generated Abelian Groups to answer the following: a. Find all abelian groups, up to isomorphism, of order p3 where p is a prime b. Use part (a) with a suitable p to list all possible abelian groups that are isomorphic to (Z2x From this list, identify the abelian group that is isomorphic to (Z2xZ8)/(1, 4))
17. (G2) Draw two non-isomorphic graphs with 4 vertices. Carefully explain how you know they are not isomorphic 17. (G2) Draw two non-isomorphic graphs with 4 vertices. Carefully explain how you know they are not isomorphic
11. a) Let R1, R2 be isomorphic rings with groups of units U1, U2 respectively. Prove that U1 and U2 are isomorphic groups. b) For 1 si k let Ri be a ring with group of units U,. Show that the group × Rk is just Ui × of units in the cartesian product R, × × Uk. 11. a) Let R1, R2 be isomorphic rings with groups of units U1, U2 respectively. Prove that U1 and U2 are isomorphic...
Theorem 4.27. Suppose G is a finite cyclic group of order n. Then G is isomorphic to Rn if n ≥ 3, S2 if n = 2, and the trivial group if n = 1. Most of the previous results have involved finite cyclic groups. What about infinite cyclic groups?