المجموعات، الزمر الجانبية والخرائط القسومية
مُقدَّم| English | المعنى |
|---|---|
| coset/ˈkɒset/ | زمرة ترافق |
| normal subgroup/ˈnɔːml ˈsʌbɡruːp/ | زمرة عادية |
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?
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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?
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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 ____.
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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.
أي إجابة تتناسب مع هذه الحالة؟
التحقق من بنية المجموعة وحساب رتب العناصر ودلائل الزمر الجزئية
Every subgroup of a nonabelian group is normal.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.