28 problems.
Chapter 9 solutions
- Upper triangular matrices are not normal in \(GL(2,\mathbb R)\) — Problem 6
- \(\langle k\rangle/\langle kn\rangle\cong\mathbb Z_n\) — Problem 8
- Isomorphic normal subgroups with non-isomorphic quotients — Problem 11
- Factor groups of abelian groups are abelian — Problem 13
- Order of \(14+\langle 8\rangle\) in \(\mathbb Z_{24}/\langle 8\rangle\) — Problem 14
- Order of \(rZ(D_6)\) in \(D_6/Z(D_6)\) — Problem 16
- Elements of \(\langle 4\rangle/\langle 20\rangle\) and its quotient of \(\mathbb Z/\langle 20\rangle\) — Problem 17
- Order of \((\mathbb Z_{10}\oplus U(10))/\langle(2,9)\rangle\) — Problem 19
- Abelian groups of order \(33\) are cyclic — Problem 21
- The quotient \((\mathbb Z\oplus\mathbb Z)/\langle(4,2)\rangle\) — Problem 23
- Identifying \((\mathbb Z_4\oplus\mathbb Z_{12})/\langle(2,2)\rangle\) — Problem 24
- Identifying \(U(32)/\{1,31\}\) — Problem 25
- Quotients of \(U(16)\) by \(\{1,15\}\) and \(\{1,9\}\) — Problem 27
- The order of \(gH\) divides the order of \(g\) — Problem 37
- Possible orders of \(a\) when \(aH\) has order \(3\) and \(|H|=10\) — Problem 38
- Isomorphisms carry normal subgroups to normal subgroups — Problem 40
- Squares in an abelian group via a subgroup and its quotient — Problem 42
- \(D_{13}\cong\operatorname{Inn}(D_{13})\) — Problem 46
- Center of a non-abelian group of order \(p^3\) — Problem 49
- Center of a group of order \(pq\) — Problem 50
- Normality of \(H/N\) in \(G/N\) — Problem 51
- Intersections of normal subgroups are normal — Problem 56
- Elements of order coprime to \(|G/H|\) lie in \(H\) — Problem 61
- \(x^m\in N\) when \(|G/N|=m\) — Problem 63
- Characteristic subgroups are normal — Problem 68
Team Problem Solutions for Ch 9
- Left cosets of a non-normal subgroup of \(A_4\) do not form a group — Team problem 10
- A normal subgroup of prime index contains \(H\) or satisfies \(NH=G\) — Team problem 47
- \(\operatorname{Aut}(G)\) is not cyclic for non-abelian \(G\) — Team problem 65