Question

Let A and B be subgroups of a group G such that G = 〈A,B〉 . $\large\tt {Prove \: that \: [A, B] ∆ G.}$​

Answer 1

Step-by-step explanation:

Answer: First, e ∈ A and e ∈ B, so e = e·e ∈ AB.

Second, if $\large a_{1} b_{1} , a_{2} , b_{2} ∈ AB$, then $\large ( a_{1} a_{2} b_{1} b_{2} = (a_{1}a_{2})(b_{1} b_{2}) ∈ AB,$ so AB is closed under the group operation.

Third, if ab ∈ AB, then (ab)−1 = a−1b−1 ∈ AB, so AB contains inverses. Notice that both closure and inverses used the given information that G be abelian.

Answer 2

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First, e ∈ A and e ∈ B, so e = e·e ∈ AB.

Second, if \large a_{1} b_{1} , a_{2} , b_{2} ∈ ABa1b1,a2,b2∈AB , then \large ( a_{1} a_{2} b_{1} b_{2} = (a_{1}a_{2})(b_{1} b_{2}) ∈ AB,(a1a2b1b2=(a1a2)(b1b2)∈AB, so AB is closed under the group operation.

Third, if ab ∈ AB, then (ab)−1 = a−1b−1 ∈ AB, so AB contains inverses. Notice that both closure and inverses used the given information that G be abelian.