I use definition of order to solve this problem
(3) (9 marks) Show that every group of order 55 has both a subgroup of order...
Show that every group of order 55 has both a subgroup of order 5 and a subgroup of order 11.
Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.
(6 points) Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.
(a) Let be a cyclic group of order . Prove that for every divisor of there is a subgroup of having order . (b) Characterize all factor groups of
ABSTRACT AGEBRA ( quotient group ) (8) Show that every subgroup of the quaternion group Q is a normal subgroup of Q, and construct the Cayley table of each quotient group. Use this to classify all homomorphic images of the group Q.
3. a. Let H be a subgroup of a commutative group G. If every element h ∈ H is a square in H (i.e., h = k 2 for some k ∈ H), and every element of G/H is a square in G/H, then every element of G is a square in G. b. Let G be a group and H a subgroup with [G : H] = 2. If g ∈ G has odd order (i.e., ord(g) is odd),...
21. Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order 6.
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element (1,2,3,4,5,6) is a cyclic group of order 6. b. Show that the subgroup generated by the element (1,3, 4, 5, 6, 7) is a cyclic group of order 6. c. Show that the subgroup generated by the element (1,2,3) is a cyclic group of order 3. d. Show that the subgroup generated by the element (6, 7) is a cyclic group of order 2....
#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.
Let G be a finite group, and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. [consider the subgroup of G] aha а