Solutions to the Algebra problems on theComprehensive
Examination of January 26, 2018
1. (25 points) Let G1 and G2 be groups, let H1 ⊆ G1 be a subgroup, and let φ : G1 → G2
be a homomorphism. The set
H2 = {φ(x) : x ∈ H1 }
is called the image of H1 under φ, sometimes denoted φ(H1 ).
Prove that H2 is a subgroup of G2 .
[This is a standard theorem in Math 350. You must actually prove it, not just quote it.]
Solution: It suffices to show that H2 is nonempty, closed under ·, and closed under −1 .
Nonempty : Let e1 be the identity in G1 . Then e1 is also the identity in H1 (e.g. e1 ∈ H1 ), so
that φ(e1 ) ∈ H2 , hence H2 is nonempty. (In fact, because φ is a homomorphism, φ(e1 ) = e2 ,
where e2 is the identity in G2 .)
Closed under · : Let z, w ∈ H2 . Then there exists x, y ∈ H1 such that z = φ(x), w = φ(y).
Since H1 is a subgroup, xy ∈ H1 , hence φ(xy) ∈ H2 . But φ is a homomorphism, so that
φ(xy) = φ(x)φ(y). Hence, zw = φ(x)φ(y) ∈ H2 , so that H2 is closed under ·.
Closed under −1 : Let z ∈ H2 . Then there exists x ∈ H1 such that z = φ(x). Since H1 is a subgroup, x−1 ∈ H1 , hence φ(x−1 ) ∈ H2 . But φ is a homomorphism, so that φ(x−1 ) = φ(x)−1 .
Hence, z −1 = φ(x)−1 ∈ H2 , so that H2 is closed under −1 .
2. (25 points) Let G be a group, let N ⊆ G be a normal subgroup, and let m ≥ 1 be an
integer. Suppose that for every element y ∈ G/N , the order of y divides m.
Prove that for all x ∈ G, we have xm ∈ N .
Solution: Let x ∈ G. Then xN ∈ G/N . Let n be the order of xN in G/N . By hypotheses,
n divides m, so there is some integer k such that nk = m. Then (xN )n = N (the identity in
G/N ), so that xm N = (xN )m = (xN )nk = ((xN )n )k = N k = N . That is, xm N = N , which
implies xm ∈ N as claimed.
3. (25 points) Consider the group S8 of permutations of the set {1, 2, 3, . . . , 8}. Let σ, τ ∈ S8
be the permutations
σ = (1, 2, 3)(4, 5, 6) and τ = (3, 5)(1, 7, 8, 4).
(a) (7 points) Write σ 2 τ as a product of disjoint cycles.
Solution: σ 2 τ = (1, 2, 3)(4, 5, 6)(1, 2, 3)(4, 5, 6)(3, 5)(1, 7, 8, 4) = (1, 7, 8, 6, 5, 2)(3, 4)
(b) (9 points) Compute the order of each of σ, τ , and σ 2 τ .
Solution: σ is given as the product of two disjoint 3-cycles, so the order of σ is lcm(3, 3) = 3.
Similarly, τ is given as the product of a 2-cycle and a 4-cycle which are disjoint, so the order
of σ is lcm(2, 4) = 4. Finally, from part (a), σ 2 τ is the product of a 6-cycle and a 2-cycle
which are disjoint, so the order of σ 2 τ is lcm(6, 2) = 6. (c) (9 points) Decide whether each of σ, τ , and σ 2 τ is an even or odd permutation;
don’t forget to justify.
Solution: σ is a product of two 3-cycles (which are both even) so is even + even = even.
τ is a product of a 2-cycle (odd) and a 4-cycle (odd), so is odd + odd = even.
σ 2 τ is a product of a 6-cycle (odd) and a 2-cycle (odd), so is odd + odd = even.
4. (25 points) Let R be a ring.
(a) (6 points) Define what it means for a subset I ⊆ R to be an ideal of R.
If you use any other technical terms like “closed,” “subring,” “group,” “subgroup,”
etc., you must fully define those terms as well.
Solution: A subset I ⊆ R is an ideal of R if
• I 6= ∅,
• x, y ∈ I implies x − y ∈ I,
• x ∈ I and r ∈ R implies xr ∈ I and rx ∈ I.
(b) (19 points) Let I, J ⊆ R be ideals, and define
I + J = {x + y : x ∈ I and y ∈ J}.
Prove that I + J is an ideal of R.
Solution: We will show I + J satisfies the three conditions given in part (a).
• Since I and J are ideals, they are non-empty. In particular, 0R ∈ I and 0R ∈ J (where
0R is the zero element in R). Then 0R + 0R = 0R ∈ I + J. Hence, I + J 6= ∅.
• Let z1 , z2 ∈ I + J. Then there exist x1 , x2 ∈ I and y1 , y2 ∈ J such that z1 = x1 + y1 and
z2 = x2 + y2 . Since I is an ideal, x1 − x2 ∈ I. Similarly, since J is an ideal, y1 − y2 ∈ J.
Thus,
z1 − z2 = (x1 + y1 ) − (x2 + y2 ) = (x1 − x2 ) + (y1 − y2 ) ∈ I + J.
• Let r ∈ R and z ∈ I + J. Then there exists x ∈ I, y ∈ J such that z = x + y. Since I
is an ideal, rx and xr are in I. Similarly, since J is an ideal, ry and yr are in J. Thus,
rz = r(x + y) = rx + ry ∈ I + J, and zr = (x + y)r = xr + yr ∈ I + J.
2
Solutions to the Algebra problems on the Comprehensive Examination 2018
of 2
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