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ANSWER 1 & 2 please. Show work for my understanding and upvote. THANK YOU!!

1. Consider the subgroups H-〈(123)〉 and K-〈(12)(34)〉 of the alternating group A123), (12) (34)). Carry out the following step

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Answer #1

All Elements of Aa ave A4=IT, (123),(132), (124), (142) , ( 1343,(143), (2 34,2 43)2 (34), (13 (24 14123) y O(A4) 124 2. In AdH we t neus Cosslg Jaking elements en aheaH then (13) (24) C) C3024 (13) (24) (123)(14 (137242 (132) (234 (13) (24)H324),142H1 (12134) CI234,(134), (234)yk Sam e 11(14.) (3)-1(14)(23))(142. ) , С 14013-8 Then we can fin we ge whese H H (12) (34(13)2lG The. Set of hose elements of G shich co m mytes with each element of G is called cenks 20) of 6 and henre zA6is ypot order 6 and is eien here CxCs is eychc on, 木so gmyps of order 6 2 Ds is not abehn hente s cox(g) is trivi 久 tab o theari) 2. i+ Same prores of modalos for 2. We have nz a is odd the. center of Dnis tivial and when ne is evon me center of Dn iWe can find is erden Centas of A abelian 1ょn-2 5, is Centor is e idenhty

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