List all cyclic subgroups of the group
Example 3.
We shall take
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Elements Of Modern Algebra
- 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 .arrow_forward9. Find all homomorphic images of the octic group.arrow_forwardIf G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.arrow_forward
- 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.arrow_forwardLet 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?arrow_forwardLet A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forward
- Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.arrow_forward9. Suppose that and are subgroups of the abelian group such that . Prove that .arrow_forwardProve part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.arrow_forward
- Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.arrow_forwardProve part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.arrow_forwardFind all subgroups of the octic group D4.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,