Groups, cosets and quotient maps
| English | 中文 | Pinyin |
|---|---|---|
| coset/ˈkɒset/ | 陪集 | péi jí |
| normal subgroup/ˈnɔːml ˈsʌbɡruːp/ | 正规子群 | zhèng guī zi qún |
A decision before an answer
- Twelve clock positions can be grouped into three classes without losing the rule for adding hours. Which positions must become equivalent?
- Your goal: Verify group structure and compute element orders and subgroup indices.
Read the relationship
- Before computing an order, check closure, associativity, an identity and an inverse for every element under the stated operation. A subset can inherit associativity yet fail closure or omit the identity. In a finite group, the order of an element is the least positive power giving the identity. In the additive group Z/nZ, it is the least positive multiple giving zero; the order of residue a is n/gcd(a,n). Lagrange's theorem says subgroup orders divide the group order. The converse is not a general existence theorem, and the order of a group is not the order of each element.
- Use kernels and images to identify quotient groups.
What is the order of 8 in the additive group Z/20Z?
The least positive k with 8k divisible by 20 is 5; equivalently 20/gcd(8,20)=5.
Use the defining rule
- A left coset gH is a translate of a subgroup H. Cosets have equal size and partition the group, so the index is |G|/|H| in a finite group. In an additive group write g+H. Membership in the same coset means the difference lies in H. A coset usually is not itself a subgroup because it may omit the identity.
- Distinguish normal subgroups from arbitrary subgroups.
A homomorphism from a finite group of order 30 has kernel of order 5. How many elements are in its image?
Fibres are kernel cosets of size 5, so the image has 30/5=6 elements regardless of the codomain size.
Check the conditions
- A homomorphism preserves the operation. Its kernel consists of elements sent to the identity, and its image consists of values actually reached. Every kernel is normal. The first isomorphism theorem identifies G/ker(phi) with im(phi); do not replace the image with the whole codomain unless the map is onto.
- Classify permutation conjugacy by cycle type.
Define phi from Z/12Z to Z/3Z by reducing residues modulo 3. It is onto and preserves addition. Its kernel H is {0,3,6,9}, so |H|=4 and the index is 12/4=3. The other cosets are {1,4,7,10} and {2,5,8,11}. Thus (Z/12Z)/H is isomorphic to Z/3Z. The element 3 in the original group has order 12/gcd(3,12)=4, not 3.
The permutation (1 2 3)(4 5) has order ____.
Disjoint cycles require a multiple of both 3 and 2 steps; lcm(3,2)=6.
Apply the task format
- Quotient multiplication is well defined only when H is normal. All subgroups of an abelian group are normal. In a nonabelian group test gHg^−1=H; a subgroup of index two is normal. For permutations compose in the stated convention, here rightmost first. Disjoint cycle lengths give the permutation order by their least common multiple. Conjugation hσh⁻¹ relabels the elements in σ’s cycles, so it preserves cycle lengths; conversely permutations with the same cycle lengths can be related by a relabelling. Thus conjugacy classes in S_n correspond to partitions of n, including fixed-point cycles. In S4 the types are 1+1+1+1, 2+1+1, 2+2, 3+1 and 4: five classes, not one class for each possible element order. The types 2+1+1 and 2+2 both have order two but are not conjugate.
- Classify permutation conjugacy by cycle type.
A quotient has one element per coset, not one per element of its kernel. A homomorphism need not be onto its stated codomain.
Which answer fits this case?
Verify group structure and compute element orders and subgroup indices
Every subgroup of a nonabelian group is normal.
In S3, conjugating the subgroup {identity,(1 2)} by (1 2 3) gives {identity,(2 3)}, a different subgroup.
Keep the distinctions
- coset 陪集 — A translate of a subgroup that forms one part of the coset partition.
- normal subgroup 正规子群 — A subgroup invariant under conjugation by every group element.
- Verify group structure and compute element orders and subgroup indices.
- Use kernels and images to identify quotient groups.
- Distinguish normal subgroups from arbitrary subgroups.
- Classify permutation conjugacy by cycle type.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.