Let G be a group of order |G|=992=2^5*31 Prove if G is Albien, find all prime po

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Definition 3.1.1. A binary operation * on a set S is a function * : S × S -> S defined on the set S × S of all ordered pairs of elements in S and taking values in S. The operation * is said to be associative if a * (b * c) = (a * b) * c for all a,b,c in S. An element e in S is called an identity element for * if a * e = a and e * a = a for all a in S. If * has an identity element e, and a is an element of S, then an element b in S is said to be an inverse element for a if a * b = e and b * a = e. Proposition 3.1.2. Let * be an associative, binary operation on a set S. (a) The operation * has at most one identity element. (b) If * has an identity element, then any element of S has at most one inverse. (c) If * has an identity element and elements a,b in S have inverses a-1 and b-1, respectively, then the inverse of a-1 exists and ( a-1 )-1 = a, and the inverse of a * b exists and ( a * b )-1 = b-1 * a-1. Definition 3.1.3. A group (G,·) is a nonempty set G together with a binary operation · on G such that the following conditions hold: (i) Closure: For all a,b in G, the element a · b is a uniquely defined element of G. (ii) Associativity: For all a,b,c in G, we have a · (b · c) = (a · b) · c. (iii) Identity: There exists an identity element e in G such that e · a = a and a · e = a for all a in G. (iv) Inverses: For each a in G there exists an inverse element a-1 in G such that a · a-1 = e and a-1 · a = e. We will usually write ab for the product a · b. Example 3.1.1. The set Q× of nonzero rational numbers, the set R× of nonzero real numbers, and the set C× of nonzero complex numbers form groups under ordinary multiplication. Definition 3.1.4. The set of all permutations of a set S is denoted by Sym(S). The set of all permutations of the set {1,2,...,n} is denoted by Sn. Proposition 3.1.5. If S is any nonempty set, then Sym(S) is a group under the operation of composition of functions. Proposition 3.1.6. (Cancellation Property for Groups) Let G be a group, and let a,b,c be elements of G. (a) If ab = ac, then b = c. (b) If ac = bc, then a = b. Proposition 3.1.7. If G is a group and a,b belong to G, then the equations ax = b and xa = b have unique solutions. Conversely, if G is a nonempty set with an associative binary operation in which the equations ax = b and xa = b have solutions for all a,b in G, then G is a group. Definition 3.1.8. A group G is said to be abelian if ab = ba for all elements a,b in G. Definition 3.1.9. A group G is said to be a finite group if the set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by |G|. Example 3.1.3. Zn is an abelian group under addition. Example 3.1.4. Zn× is an abelian group under multiplication. Its order is given by the value (n) of Euler's phi-function. Definition 3.1.10. The set of all invertible n × n matrices with entries in R is called the general linear group of degree n over the real numbers, and is denoted by GLn(R). Proposition 3.1.11. The set GLn(R) forms a group under matrix multiplication. § 3.1 Definition of a Group: Solved problems The definitions in this section provide the language you will be working with, and you simply must know this language. Loosely, a group is a set on which it is possible to define a binary operation that is associative, has an identity element, and has inverses for each of its elements. The precise statement is given in Definition 3.1.3; you must pay careful attention to each part, especially the quantifiers ("for all", "for each", "there exists"), which must be stated in exactly the right order. From one point of view, the axioms for a group give us just what we need to work with equations involving the operation in the group. For example, one of the rules you are used to says that you can multiply both sides of an equation by the same value, and the equation will still hold. This still works for the operation in a group, since if x and y are elements of a group G, and x = y, then a ·: x = a · y, for any element a in G. This is a part of the guarantee that comes with the definition of a binary operation. It is important to note that on both sides of the equation, a is multiplied on the left. We could also guarantee that x · a = y · a, but we can't guarantee that a · x = y · a, since the operation in the group may not satisfy the commutative law. The existence of inverses allows cancellation (see Proposition 3.1.6 for the precise statement). Remember that in a group there is no mention of division, so whenever you are tempted to write a ÷ b or a / b, you must write a · b-1 or b-1 · a. If you are careful about the side on which you multiply, and don't fall victim to the temptation to divide, you can be pretty safe in doing the familiar things to an equation that involves elements of a group. Understanding and remembering the definitions will give you one level of understanding. The next level comes from knowing some good examples. The third level of understanding comes from using the definitions to prove various facts about groups. In the study of finite groups, the most important examples come from groups of matrices. I should still mention that the original motivation for studying groups came from studying sets of permutations, and so the symmetric group Sn still has an important role to play. 22. Use the dot product to define a multiplication on R3. Does this make R3 into a group? Solution 23. For vectors (a1,a2,a3) and (b1,b2,b3) in R3, the cross product is defined by (a1,a2,a3) × (b1,b2,b3) = (a2b3-b3a2, a3b1-a1b3, a1b2-a2b1). Is R3 a group under this multiplication? Solution 24. On the set G = Q× of nonzero rational numbers, define a new multiplication by a * b = ab/2, for all a,b in G.

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