Question

Adventures in Algebra VII: This is completely normal. 1 Let n be a positive even integer. Recall that the dihedral group D is generated by r ands subject to r" = s? = c and rs = sr-1. Show that (-2) is a normal subgroup of Dm

proof should involve r^(2i) for some i

Answer 1

Soeh of Dn) = 2h claim : -cr²y is normal Dna first we }<0,8) ghe = 8² 88 = 8873 3 (۲) subgroup of G Is Two subgroup of Dn. obfcrue That Her> 8h e 3 set H be cyclic subgroup generated byar.> Then 0(4) = 10l4l=h And HSG Also we know That het Gibe any group And N be any of N in G if. Index Then N is normal inG.. Index 0(H) above subgroup Ha is hormal subgroup of G c.c txt G. xhulth Now K= (Y? >= ?,? - c.] <p> <P> And K be subgroup of Hol easily Proot by Step h. Index of h Hence H xHatch thth. nth, ΟΥ One subgroup Test ) We know Now (R

We'll Result N be hormal Subgroup of y. It Nis cyclic. Then Every subgroup of N. is also hormal in Go use This result H is cyclic subgroup of G. And Also k is generated by=c82> Then K SH Hence k is hormal subgroup of G. <82> is normal subgroup of xkx'ck v mehkek x K x GK Proof And ΟΥ That is Hunce
​​​​​​