BOOK: A first course in Abstract Algebra Author: John B. Fraleigh 7th edition IS
ID: 2939402 • Letter: B
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BOOK: A first course in Abstract Algebra Author: John B. Fraleigh 7th edition ISBN: 0-201-76390-7 CH. 14 P.142 Q.2 Question: Find the order of the given factor group. (Z4 x Z12) / ( <2>x<2>) I found that (Z4 x Z12) has 48 elements. e=(0,0)Identity. I also found that H= <2,2> =(2,2),(3,3),(0,4),(1,5),(2,6),(3,7),(0,8),(1,9),(2,10),(3,11),(0,0).H has 11 elements, but it suppose to have 12 elements. Because, G/H = 48/12 = 4 this is the order of the givenfactor group. Question: a) Do I have tofind the inverse of all of the 48elements? b) Do you know which one is the12th element? Thanks. BOOK: A first course in Abstract Algebra Author: John B. Fraleigh 7th edition ISBN: 0-201-76390-7 CH. 14 P.142 Q.2 Question: Find the order of the given factor group. (Z4 x Z12) / ( <2>x<2>) I found that (Z4 x Z12) has 48 elements. e=(0,0)Identity. I also found that H= <2,2> =(2,2),(3,3),(0,4),(1,5),(2,6),(3,7),(0,8),(1,9),(2,10),(3,11),(0,0).H has 11 elements, but it suppose to have 12 elements. Because, G/H = 48/12 = 4 this is the order of the givenfactor group. Question: a) Do I have tofind the inverse of all of the 48elements? b) Do you know which one is the12th element? Thanks.Explanation / Answer
( <2>x<2>) is the cyclic subgroup generated by<2>,<2> in Z4 x Z12 ( <2>x<2>) = { (2,2) , ( 0,4),(2,6),(0,8),(2,10),(0,0) }= H see that Z4 and Z12are the additive groups and the operation onZ4 x Z12 must be like ( a,b) * ( c,d)= ( a+b mod 4, c+d mod 12) see that H has 6 elements and Z4 x Z12has 48 elements and consequently, Z4 x Z12/ (<2>x<2>) has 12 elements. in your approach, you felt ( <2>x<2>) itself has12 elements which is not correct. o.k.? any more infermation you require , i can provideyou. but let me know about it. thank you. ( <2>x<2>) = { (2,2) , ( 0,4),(2,6),(0,8),(2,10),(0,0) }= H see that Z4 and Z12are the additive groups and the operation onZ4 x Z12 must be like ( a,b) * ( c,d)= ( a+b mod 4, c+d mod 12) see that H has 6 elements and Z4 x Z12has 48 elements and consequently, Z4 x Z12/ (<2>x<2>) has 12 elements. in your approach, you felt ( <2>x<2>) itself has12 elements which is not correct. o.k.? any more infermation you require , i can provideyou. but let me know about it. thank you.Related Questions
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