True or False Label each of the following statements as either true or false. ✓1. aHHaØwhere H is any subgroup of a group G and a Є G. 2. Let H be any subgroup of a group G. Then H is a left coset of H in G. 3. Let H be any subgroup of a group G and a E G. Then aH = Ha. 4. The elements of G can be separated into mutually disjoint subsets using either left cosets or right cosets of a subgroup H of G. 5. The order of an element of a finite group divides the order of the group. 6. The order of any subgroup of a finite group divides the order of the group. 7. Let H be a subgroup of a finite group G. The index of H in G must divide the order of G. 8. Every left coset of a group G is a subgroup of G.

True or False Label each of the following statements as either true or false. ✓1. aHHaØwhere H is any subgroup of a group G and a Є G. 2. Let H be any subgroup of a group G. Then H is a left coset of H in G. 3. Let H be any subgroup of a group G and a E G. Then aH = Ha. 4. The elements of G can be separated into mutually disjoint subsets using either left cosets or right cosets of a subgroup H of G. 5. The order of an element of a finite group divides the order of the group. 6. The order of any subgroup of a finite group divides the order of the group. 7. Let H be a subgroup of a finite group G. The index of H in G must divide the order of G. 8. Every left coset of a group G is a subgroup of G.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

5. Prove that the intersection of two subgroups is always a subgroup. i.e. If G is a group and H and K are subgroups of G, then H N K is also a subgroup of G.
Question 6. Given a group (G, *) and a nonempty set S. Let GS denote the set of all mappings from the set S to the set G. Find an operation on GS that will yield a group. Show that your choice of operation is correct.
4. Let a be an element of a group. If |a| = 10, what are the possibilities for a? 5. List all the elements of Z100 that have order 10.
Let H be a subgroup of G and let a, be G. If Ha = Hb, then * aH = b-H a-1H = b-1H Ha = Hb Ha-1 = Hb-1 Ha-1 = Hb-1 None of the choices
If a and b are arbitrary distinct elements of a group G and H is any subgroup of G, then Ha=Hb⇔ab\power{-1}∈H aH=bH⇔b\power{-1}a∈H.
Let H be a subgroup of a group G and a, bEG. ThenbE aH if and only if * O None of these O ab EH O ab-1 EH O a-1b e H