Let A be the abelian group with generators a, b, c, d and relations 2a 4b + с, 4c-d-2b and a + b + c + d-0. Write A as...

Question

Question

Let A be the abelian group with generators a, b, c, d and relations 2a 4b + с, 4c-d-2b and a + b + c + d-0. Write A as the ca

Answer 1

Answer #1

The given relations are

$c=2a-4b----------------(i)$

$d=2b+4c----------------(ii)$

$a+b+c+d=0--------------(iii)$

Substituting the values of c and d in (iii) we get

$=> 15b-5a= 0$

Substituting the value of a in (i) we get, $c=2b$

Substituting the value of c in (ii) we get $d = 106$

Thus we get $a, b,c,d E< b >$ , where $<\,>$ denotes the cyclic group generated by.

Since A is an abelian generated by a,b,c,d then $A\subset <b>$ , and since $b\in A=> <b>\subset A$ . Thus we get $A= <b>$ . A cyclic group.

Feel free to comment if you have doubts. Cheers!

Answer 2

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