(1 point) Let x be an element of order 26 in a group G (not necessarily...
Order and Cyclic Subgroups: Problem 5 Previous Problem Problem List Next Problem (1 point) Let x be an element of order 91 in a group G (not necessarily cyclic, finite, or Abelian). How many distinct subgroups of G are contained in (x)?
Only 2 and 3 1.) Let G be a finite G be a finite group of order 125, 1. e. 161-125 with the identity elemente. Assume that contains an element a with a 25 t e, Show that is cyclic 2. Solve the system of congruence.. 5x = 17 (mod 12) x = 13 mod 19) 3.) Let G be an abelian. group Let it be a subgroup o G. Show that alt -Ha for any a EG
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2
Problem 1. Let G be a finite group and f : G → G a group automorphism ( isomorphism for G to G) of order 2 (i.e. f(f(x)) = x), and f has no nontrivial fixed points (i.e. f(x) = x if and only if x = 1). Prove that G is an abelian group of odd order.
please show step by step solution with a clear explanation! 2. Let G be a group of order 21. Use Lagrange's Theorem or its consequences discussed in class to solve the following problems: (a) List all the possible orders of subgroups of G. (Don't forget the trivial subgroups.) (b) Show that every proper subgroup of G is cyclic. (c) List all the possible orders of elements of G? (Don't forget the identity element.) (d) Assume that G is abelian, so...
Let G be a cyclic group of order 30. Find all the subgroups of G. Write the lattice of subgroups. Justif your answer and cite the theorems that allow you to determine such lattice.
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than G has exactly two subgroups. What are they? 1 b) Let G be a group and H a subgroup of G. Let x ∈ G. Proof that if for a, b ∈ H and ax = b then x ∈ H. (If you use any group axioms, show them)
14. Suppose that G is a group in which {1} and G are the only subgroups. Show that G is finite and, in fact, is cyclic of order 1 or a prime.
Please answer all the four subquestions. Thank you! 2. In this problem, we will prove the following result: fG is a group of order 35, then G is isomorphic to Z3 We will proceed by contrd cuon, so throughout the ollowing questions assume hat s grou o or ㎢ 3 hat s not cyc ić. M os hese uuestions can bc le nuc endent 1. Show that every element of G except the identity has order 5 or 7. Let...