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  3. let d10 denote the dihedral group of the hexagon.

Let D10 denote the dihedral group of the hexagon. Thus, D10 is generated by r and...

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Advance Math

Let D10 denote the dihedral group of the hexagon.Thus, D10 is generated by r and f withr10=f2=1 andfr=r-1f=r9f

(a) Show that D10 has a subgroups N and M such that
i. N ?= D5 (isomorphic to D5)
ii. M is a cyclic subgroup group of order 2
iii. N ? M = {e}
iv. N and M are each normal in D10
v. Every element in g ? D10 is a product g = nm of elements n ? Nand m ? M.

(b) Prove that if G is any group with normal subgroups N and Msuch that N ? M = {e}, then
nm = mn for all n ? N and m ? M.

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