I need to prove that if $g$ and $h$ commute and their orders are coprime, then $|gh| = |g||h|$, that is, the order of their product is the multiple of their orde...

2018年8月29日 · Given the following definition of normal subgroup A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$ I've tried to show that $H\mathrel …

Formally a subgroup is normal if every left coset containing g is equal to its right coset containing g. rm g. This means that for any g 2 G we don't necessarily get gh = hg but at worst we get gh …

If [G : H] = 2, prove that gH = Hg. Solution. Since there are only two left cosets of H, which are disjoint, and one of them is H itself, the left cosets are H and G H. The same holds for the right …

2021年5月10日 · In nearly all proofs for the "subgroup of index 2 is normal" statement, there is a sentence that reads more or less: if $g\in H\subseteq G$ then $$gH=H=Hg,$$ but why is this …

If g is in H, then gH = H = Hg. If g does not belong to H, then gH is the left coset that is different from H and Hg is the right coset that is different from H and so gH = Hg.

而 H 的不同陪集只有两个： H 与 G-H 。 故 gH=G-H 。 同理地，可以证明 Hg=G-H ，故 gH=Hg 。 如果一个子群所含元素个数恰好是群中元素个数的一半，则其必为正规的。

A subgroup H H of a group G G is normal in G if gH = Hg g H = H g for all g ∈ G. g ∈ G. That is, a normal subgroup of a group G G is one in which the right and left cosets are precisely the …

设g及h 是群G中的元素，他们的交换子是g -1 h -1 gh，常记为 [g, h]。 只有当g和h符合交换律（即gh = hg）时它们的交换子才是这个群的单位元。 一个群G的全部交换子生成的子群叫做群G的 …

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