7. Let G be a group. Then Z(G) is always of smaller size than G. *True *False
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- Write 20 as the direct sum of two of its nontrivial subgroups.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
- Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .4. Prove that the special linear group is a normal subgroup of the general linear group .Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.
- 3. Suppose that a group G contains elements of orders 1, 2, 3, 4, ..., 8, 9. What is the minimum order of group G? Explain your answer.9. If G is a group then Z(G)Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your answer.Let (G, *) and (H,) be groups. If group (G, *) is cyclic and G = H, then (H,) is also cyclic. Select one: O True O FalseIf G and G' are both groups then G' ∩ G is a group. True or False then why3. Verify Test the properties of a group : Let G denote a set of all ordered pairs of real-valued numbers with nonzero first component. Define the operation * by the rule (a, b) * (c, d) = (ac, be + d), V(a, b), (c, d) e G %3DSEE MORE QUESTIONS