Determine whether each of the following statemests statementis True or False - W

Question

Determine whether each of the following statemests statementis True or False - Why? 1) Let G be a group, a, b elements of G, o(a) = 2, o(b) =5. Then o(ab) = 10. 2) If n divides ¦G¦, then we can find aelement of G such that o(a) = n 3) If G is an abelian group, then any subgroup of G isnormal Determine whether each of the following statemests statementis True or False - Why? 1) Let G be a group, a, b elements of G, o(a) = 2, o(b) =5. Then o(ab) = 10. 2) If n divides ¦G¦, then we can find aelement of G such that o(a) = n 3) If G is an abelian group, then any subgroup of G isnormal 3) If G is an abelian group, then any subgroup of G isnormal

Explanation / Answer

1) If G were abelian, that would be true. But look atthe permutation group of 5 elements, and let a be a 2-cycle and bbe a 5-cycle. Whichever a and b you pick, you'll have acounterexample. (There are lots of possible orders for ab in thatscenario, but 10 isn't one of the possibilities.) 2) If G were cyclic, that would be true. But you canconstruct a group of 4 elements where every element has order2. Then n = 4 is a counterexample. 3) True. That's because if you conjugate an element c byanother element b, the result is just c.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

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