Modern Algebra, Group Theory (XXVI): Subgroups of a Group: If H and K are subgroups of G, then so is H∩K a subgroup of G.

Modern Algebra Group Theory (XXVI) Subgroups of a Group If H and K are subgroups of G, then so is H∩K a subgroup of G.

Question about group theory, cardinality and isomorphic

I would like to know how to identify and prove the cardinality of sets and how to identify isomorphic. (See attached file for full problem description) --- Group Theory: a. If S and T are sets then let TS denote the set of all functions from S to T. Prove that the cardinality of TSxU equals the cardinality of (TS)U b. Consider the groups Z3 x Z3 and Z9. These are each "integer groups" of order 9. Are they isomorphic or not? Give an explicit reason. Z- integer ---

Group Structure, Order of two groups

I have questions about constructing a group structure, how to identify the order of a paired group when they have different orders and method of figuring out the group identity and the inverse of a pair that contained in the paired group. --- If G and H are groups then explain how to equip G x H with a group structure. If G has order k and H has order m then what is the order of G x H? What is the group identity and the inverse of an element (g, h) Є G x H. ---

For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH.Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of Subgroups of a Group For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH. Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH.Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of Subgroups of a Group For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH. Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.