Lemma Let B Be An Element Of A Group G And Let N 0 I J Ne Z Assume That The Powers E Bo B B B B B Are 1 (41.82 KiB) Viewed 65 times
Lemma Let B Be An Element Of A Group G And Let N 0 I J Ne Z Assume That The Powers E Bo B B B B B Are 2 (20.26 KiB) Viewed 65 times
Lemma: Let b be an element of a group G and let n 0<i<j≤ne Z*. Assume that the powers e = bº, b', b², b, b, ...., b" are not all distinct, that is, there are integers i and j so that b¹ = b for 0≤i<j≤n. Then there is a positive integer k ≤ n such that b*= e. Definition: The number k in the lemma is called the order of the element b. Every element in a group has an order, and they are not all necessarily the same order. (So an element in a group of 6 elements could have order 2, and another element might have order 6. But an element cannot have order 7 in a group of order 6.) a) Find the order of all elements of Z3. q Find the order of all elements of Z4. b)
d) Find the order of all elements of V (viergruppe). Find the order of all elements of D3. (dihedral group on the transformation of the equilateral triangle)
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