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is normal in ; for , , while for
Statement
For every natural , is a normal subgroup of . If , then . If or , then .
Facts & Assumptions
Given: A natural , the sign homomorphism on , and the alternating group .
is the kernel of sign (The alternating group of even permutations).
The kernel of every group homomorphism is a normal subgroup (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).
The symmetric group of an -element set has elements (A finite set with has exactly bijections onto itself, and bijections onto any set of the same cardinality).
Sign is a homomorphism whose image is exactly when , and is for (The sign is a homomorphism , surjective exactly when ).
The cardinality of a disjoint union of two finite sets is the sum of their cardinalities (The sum rule: a finite disjoint union is finite with and , and a sum over a finite index set splits along a partition).
Proof
By [L1] and [L2], is a normal subgroup of .
Suppose . By [L4], choose with . Left multiplication is a bijection from the even fibre of sign to the odd fibre, with inverse left multiplication by , because [L4] gives .
The even and odd fibres are disjoint and have union , and step 2.1 gives them equal finite cardinality. Thus [L5] and [L3] give .
If or , [L4] says that sign has image , so its kernel is all of and .
Depends on
- The alternating group $A_n=\ker(\operatorname{sgn})$ of even permutations
- The sign is a homomorphism $S_n\to\{+1,-1\}$, surjective exactly when $n\ge 2$
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
- A finite set $A$ with $\lvert A\rvert = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality
- The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
- The sum rule: a finite disjoint union is finite with $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$, and a sum over a finite index set splits along a partition
- The cardinality $\lvert A\rvert$ of a finite set
Used by
- For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
- A₄ consists of the identity, eight 3-cycles, and three products of disjoint transpositions Example
- The conjugacy classes of A₅: sizes 1,20,15,12,12 and the split 5-cycles Example
- V₄={1,(12)(34),(13)(24),(14)(23)} is a proper nontrivial normal subgroup of A₄ Example
- FALSE: Aₙ is simple for every n≥4 False statement
- FALSE: every nontrivial normal subgroup of a faithful primitive group is regular False statement
- The cycle index of Aₙ is the parity-filtered symmetric-group sum Theorem
Dependency tree · two levels
45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- T. W. Judson, Abstract Algebra: Theory and Applications, §5.1, Theorem 5.7 and Proposition 5.8 (standard reference, not scraped)