need help with #2 and extra credit on homework. Thanks Let a1 epsilon G1 be an e

Question

need help with #2 and extra credit on homework. Thanks

Let a1 epsilon G1 be an element of the group G (with.) having order 6. Now let a2 epsilon G2 (also with .) have order 10. Find ord((alpha 1, alpha 2)) as an element of G1 times G2. Justify your answer. Recall that two integers m, n epsilon Z are said to be relative prime if gcd(m, n) = 1 This means that (m) + (n) = (1) = Z. So 1 epsilon (m) + (n) and we can write 1 = sm + tn for some s,t epsilon Z. Use this fact to show that if a epsilon G (G a group with .) and if am = c and an = c where gcd(m, n) = 1 then a = c, Hint: Use the fact that a = a1. Extra Credit Let a epsilon G (G a group with .), Suppose that gcd(m, n) = 1 for two integers m and n and that am = e. Prove then that a has an n-th root in G, i.e. that there is a b epsilon G with bn = a.

Explanation / Answer

ORDER OF A1 IN G1 = 6.......THAT IS ....A1^6=E.............. WHERE E IS IDENTITY ELEMENT IN G1.......................................................1

ORDER OF A2 IN G2 = 10.......THAT IS ....A2^6=E' ................ WHERE E' IS IDENTITY ELEMENT IN G2.............................................................2

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.