Nhóm, lớp đồng dư và ánh xạ thương

Grouping the clock

• Twelve clock hours can be sorted into three classes — $\{0,3,6,9\}$, $\{1,4,7,10\}$, $\{2,5,8,11\}$ — and adding hours still works class by class.
• That is a quotient group. GRE group theory tests orders, cosets 陪集, Lagrange, kernels and normal subgroups 正规子群.

Orders and Lagrange

• The order of $g$ is the least $k > 0$ with $g^k = e$. In $\mathbb Z/n\mathbb Z$, the order of $a$ is $\dfrac{n}{\gcd(a,n)}$: in $\mathbb Z/12$, the element 3 has order 4.
• Lagrange: subgroup orders divide $|G|$. The converse fails in general: $A_4$ (order 12) has no subgroup of order 6.

Cosets and index 指数

• $g + H$ is a translate of $H$. Cosets have equal size and partition $G$, so the index is $|G|/|H|$.
• $a, b$ share a coset exactly when $a - b \in H$. A coset other than $H$ is not a subgroup: it misses the identity.

Kernels and quotients

• $\ker\varphi$ is normal and $G/\ker\varphi \cong \operatorname{im}\varphi$ — the image, which equals the codomain only if $\varphi$ is onto.

Normality and permutations

• $G/H$ is a group only if $H$ is normal: $gHg^{-1} = H$. Abelian subgroups and index-2 subgroups always are.
• In $S_n$: order = lcm of disjoint cycle lengths; $(1\,2)(3\,4\,5)$ has order 6. Conjugate permutations have the same cycle type, so $S_4$ has 5 conjugacy classes (partitions of 4).

Recap