Update:

John Brooks drew a **elementary algebra questions and solutions** for a Somes Receptive to advice as well as I purchased my set in November of 2010 as well as proposed growth in 2011?

Aogebra little people have a grassed area to work in others might have large sheds as well as even entrance to the warehouse. Glue is only not self leveling.

In this essayA sleet right here in H2O World; aka: Horse opera WA.

My son could practice the types of questions and have an understanding of what he was going to be asked and confidence going into the test.

Answer and Explanation The correct answer is A. To solve this problem, apply the equation rules of operations. Simplify the equation by reducing and moving like-terms: According to the calculations, the correct answer is A. If you chose any other answer, review your calculations. Every day that Jim eats something other than oatmeal for breakfast, he gives back 5 smiley stickers. The number of days that Jim ate oatmeal for breakfast this year is represented by c , and the number of days he ate something other than oatmeal for breakfast this year is represented by d.

Which of the following represents the number of stickers Jim holds at the end of the year? Scroll down to reveal the answer and explanation. Answer and Explanation The correct answer is B. Write down all the information given in the question. If you chose any other answer, review the setup of your equation. Answer and Explanation The correct answer is D.

To simplify the expression, the numerator and the denominator should be factored independently. As part of the quadratic expression definition, the equation equals zero; thus, you can manipulate it: Rewrite the equation: Therefore, the correct answer is D. If you chose any other answer, review your calculations and the rules of factoring. Related Links. Let be a group and suppose that is a subgroup of such that Show that is normal in.

Choose Since we have Now, suppose, to the contrary, that is not normal. So there exist such that So and Therefore there exist such that and But then, eliminating we get contradiction. Are subgroups of prime index always normal? Of course not. For example, let be any prime number, and let be the dihedral group of order i. Problem 3 T. Let be a group, and let be a prime number. Suppose that has a subgroup such that Show that the following statements are equivalent.

Since is normal and the set is a group of order and hence for all i. Let and suppose, to the contrary, that for some integer Then, since is prime, and hence there exist integers such that But then we get contradiction. Suppose, to the contrary, that is not normal. Hence there exist and such that Thus, by iii , for all integers and hence we have distinct cosets Thus, since we get So and hence for some and some integer Therefore. Remark 1. Problem 4 J. Let be a prime number. Show that if is a finite group and is a subgroup of such that and then is normal in.

Case 1 : In this case, and clearly Also, since divides and we also have Thus, by our induction hypothesis, is normal in and hence is normal in. Case 2 : In this case, and thus, since we get that for some integer with Since we have and also. Since is normal, is a subgroup of and thus, by must divide and that is possible only if Thus and hence which gives because So is normal in and the solution is complete. Remark 2. Note that Problem 1 follows from Problem 4.

Let be a vector space over a field of characteristic Let be two linear transformations such that where is the identity map. Show that. If then. Let So and hence which give. Recall that an element of a group is called torsion if it has a finite order and a group is called torsion if every element of the group is torsion.

A non-trivial group is called torsion-free if the only torsion element of the group is the identity element. Clearly every finite group is torsion and every torsion-free group is infinite. Let be a torsion-free group and suppose that has a cyclic subgroup of finite index. Show that is cyclic. Theorem Schur. Let be a group with the center and the commutator subgroup If is finite, then is finite too. See this post! Let be a torsion-free group. If is finite, then is abelian.

But is torsion-free and so i. I give the solution in several steps. Claim 1. We may and will assume that is normal in Also,. Claim 2. If is abelian, then is cyclic. Let Then the map defined by is a group homomorphism and, since is torsion-free, is injective. Thus is isomorphic to a subgroup of and hence it is cyclic because is cyclic.

Claim 3. Let be the centralizer of in Then is cyclic. Thus by the above Corollary, is abelian and hence cyclic, by Claim 2. Claim 4. If has a cyclic subgroup of index then is cyclic. Thus either or If then which gives contradiction.

Hence and therefore i. Claim 5. If then again is cyclic, by Claim 3 and Claim 4. The proof of the claim and the solution is now complete. Problem V. Find all functions that satisfy the following functional equation. First see that such functions satisfy Now suppose that satisfies and let. By we have the identity matrix, and so. Put the above in which holds by to get. Recall that in a ring an element is said to be a left zero divisor if for some non-zero element Similarly, is a right zero divisor if for some non-zero element A zero divisor of is an element which is both a left and a right zero divisor.

Proposition 1. Let be a ring, and let be the set of left zero-divisors of If is finite and then is finite and. Since there exists such that Let. If then and so because Thus and hence. Now clearly is a subgroup of the abelian group and so we may consider the group and the map. Then is well-defined and one-to-one because clearly for all and if then if and only if if and only if if and only if if and only if Thus.

It now follows from that. Example 1. The equality in the above proposition is possible, i. Example 2. To start another set of problems, press "reset". Math Problems and Online Self Tests. Basic Rules and Properties of Algebra. Free Mathematics Tutorials.

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