? asked in 科學數學 · 1 decade ago

commutator subgroup

請教一下這題

If G is a group, H is a subgroup which is generated by all elements

xy(x^-1)(y^-1) with x,y belong to G.

prove that H is normal.

感謝,贈送五點.:)

1 Answer

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  • 1 decade ago
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    g^(-1)xy(x^-1)(y^-1)g

    = [g^(-1)xg][g^(-1)yg][g^(-1)x^(-1)g][g^(-1)y^(-1)g]

    = pqp^(-1)q^(-1)

    其中 p= g^(-1)xg, q=g^(-1)yg,

    =>p^(-1)= g^(-1)x^(-1)g, q^(-1)= g^(-1)y^(-1)g

    so, g^(-1)Hg contained in H

    H is normal subgroup of G

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