左右陪集等价性及拉格朗日定理相关性质技术问询

Great question—these are critical foundational concepts in group theory, so let’s unpack each part clearly:

1. 左、右陪集的数量是否相同？ #

Absolutely! For any subgroup ( H ) of a group ( G ), the number of distinct left cosets (denoted ( [G:H] ), called the index of ( H ) in ( G )) is exactly equal to the number of distinct right cosets.

The key reason is there’s a bijective correspondence between left and right cosets: map each left coset ( gH ) to the right coset ( Hg^{-1} ). This mapping is well-defined (if ( gH = kH ), then ( Hg^{-1} = Hk^{-1} )) and invertible (its inverse maps ( Hg ) to ( g^{-1}H )), so it perfectly pairs every left coset with a unique right coset and vice versa.

2. 每个左、右陪集的元素个数是否一致？ #

Yes, every left coset and every right coset has exactly the same size as the subgroup ( H ) itself.

Take a left coset ( gH ): define a function ( f: H \to gH ) by ( f(h) = gh ). This function is injective (if ( gh_1 = gh_2 ), multiply both sides by ( g^{-1} ) to get ( h_1 = h_2 )) and surjective (every element of ( gH ) is ( gh ) for some ( h \in H )), so it’s a bijection. The exact same logic applies to right cosets ( Hg ): the function ( f(h) = hg ) is a bijection between ( H ) and ( Hg ).

This is why Lagrange’s theorem works for both left and right cosets—since all cosets are the same size, dividing ( |G| ) by ( |H| ) gives the number of cosets (the index).

3. 非正规群中，是不是每个左陪集都和对应的右陪集不相等？ #

Not exactly—let’s clarify this common misconception:

A subgroup ( H ) is normal in ( G ) if and only if ( gH = Hg ) for every ( g \in G ). So if ( H ) is not normal, that means there exists at least one element ( g \in G ) where ( gH \neq Hg ). It does not mean that all left-right coset pairs are unequal.

For example, take the symmetric group ( S_3 ) (permutations of 3 elements) and the subgroup ( H = { e, (1\ 2) } ):

• For ( g = e ), ( eH = He = H ) (they’re identical).
• For ( g = (1\ 3) ), the left coset ( (1\ 3)H = { (1\ 3), (1\ 3\ 2) } ), while the right coset ( H(1\ 3) = { (1\ 3), (1\ 2\ 3) } )—these are clearly not equal.

Non-normality only guarantees that some left-right coset pairs differ, not all.

内容的提问来源于stack exchange，提问作者gbox