5. Prove that a group of order 63 must have an element of order 3.
5. Prove that the center of a group is the intersection of all the centralizers. 6. Is the center of a group abelian? For any group element g is the centralizer of g abelian? 7. Prove that a group of even order must have an element x so that x= 2.
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2
3 (Due 8/7) Prove that every element of a group has a unique inverse. (Due 8/7) Let (G, *) be a group and let a be an element of G with inverse d'. Prove that the function f(x) = a*r*d' is a permutation of G.
What is the order of an element of a group, find the order of 4 (mod 7). And what is a primitive element mod n, is 5 a primitive element modulo 19? Verify or refute it.
In the problems below, give the
order of the element in the indicated
factor group.
(a) in
(b) in
(c) in
(5)(20 points) In the problems below, give the order of the element in the indicated factor group. (a) (1, 2)+ < (1,1) > in Z3 x Z6/ < (1,1) >. (b) (3, 2)+ < (4,4) > in Z6 * Z8/ < (4,4) >. (c) 26+ < 12 > in Z60/ <12>.
3. Find the order of each element of the multiplicative group (Z/12Z)*.
Prove or disprove There exists nonabelian group G (order of group G is 219 3 x 73)
Please answer all the four subquestions. Thank you!
2. In this problem, we will prove the following result: fG is a group of order 35, then G is isomorphic to Z3 We will proceed by contrd cuon, so throughout the ollowing questions assume hat s grou o or ㎢ 3 hat s not cyc ić. M os hese uuestions can bc le nuc endent 1. Show that every element of G except the identity has order 5 or 7. Let...
Problem 3 () (2 marka) Prove that the group R and the circle group St are not isomsorphic to each other. Hind เตบ๐s fad element of order 2 m S., Hou about RV (a)(2marks) Let n 2 be an integer, give an escample (including explanatlon) of a group G and a subgroup FH with IG: H-nsuch that H is not normal in G. (iii) (S marks) Let G-16:l : a,b,c ER, a 7.0, eyh 아 You are given that G...
#11
11. If a group G has exactly one subgroup H of order k, prove that H is normal in G. Define the centralizer of an element g in a group G to be the set 12.