Question: Let G be a group. For any a e G define the function da : G - G by d.(x) = ara-1. (a) Show that

Let G be a group. For any a e G define the function da : G - G by d.(x) = ara-1. (a) Show that da is an automorphism of G (to show that o. is a bijection, show that the function (a-1 is its inverse). An automorphism of the form da is called an inner automorphism. (b) Show that da is the identity map if and only if a lies in the center of G

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Solve 43. Let G be a group and let g E G. If z E Z(G), show that the inner automorphism induced by g is the same as the inner automorphism induced by zg (that is, that the mappings , and o, are...

Question 6 Recall the symmetries of the square, consisting of: (a) Let H be the set of those symmetries which send the line ac to itself. That is, a H = (0 ED : o(fa,c}) = {a,c}}. Show that H is a...

Please solve this question 47. Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G). 48. Combine the results of Exercises 45 and 47 into a single "if and only...

Suppose that g and h induce the same inner automorphism of a group G. Prove that h-lg E Z(G). 48. Combine the results of Exercises 45 and 47 into a single "if and only if theorem. 49. If a and B are...

Kindly solve it urgently 47. 49. 50. 51. 52. 53. 54. 55. 56. ban-.111 . Suppose that g and h induce the same inner automorphism of a group G. Prove that h'lg E 2(6). . Combine the results of...

HW 12, MAT 312/AMS 351: SPRING 2017 Problem 1. (1) Set H = {4a + 6b | a, b Z} Z. Show that H is a subgroup of Z. (2) Is H cyclic? Problem 2. Write down all the cyclic subgroups of the following...

This is a Group Theory question. The QUESTION and SOLUTION is shown below. My question relates to part d) only. I have three questions regarding SOLUTION shown for part d): I would like to know why...

11. Write out the cycle decomposition of the eight permutations in $4 corresponding to the elements of Dg given by the action of Dg on the vertices of a square (where the vertices of the square are...

Solve action. 2. Let (F, +, .) be a field. Let G = (F, .) the group. Let V be any vector space over F (Refer chapter on Vector spaces). Define * by a * v = a.v, a e F, v E V, where . is scalar...

April 2011 WEATH3021 Group Theory Midterm test QUESTION 1 (a) Let G be a group and / be a subgroup of G. Define what is meant by a Normal subgroup of G and give one equivalent statement (3) (b) Let H...

Requirements are to be planned for component C5 in product P1. Required deliveries for P1 are given, and the product structure for P1. Assembly lead time for products and subassemblies (P and S...

A firm is considering the following investment project: Year Before tax Cash Flow (thousands) 0......................................................-$1000...

A perfect hedge eliminates the mismatch between the net assets and net liabilities in every currency.

Part 1 What is the total stockholders' equity of Entity A based on the following account balances? Common Stock $950,000 Paid-In Capital in Excess of Par 50,000 Retained Earnings 175,000 Treasury...