Abstract Algebra, Lec 6A: Properties of Mod n Addition, Order, Subgroups, (One Step) Subgroup Test

(0:00) Lecture plans.
(0:54) Mod n is really a function from the integers to {0,1,2,...,n-1}.
(3:02) The key properties for addition mod n to "work".
(3:52) Careful proof of Exercise #7 from Chapter 0 (a mod n = b mod n iff n divides a-b).
(15:20) Idea of proof of Exercise #9 from Chapter 0 (a mod n = a' and b mod n = b' implies (a+b) mod n = (a' + b') mod n.
(20:54) Review order of a group and order of an element in a group.
(23:35) Subgroups (emphasize the importance of using the same operation by noting that, for example, Z4 is not a subgroup of Z5), discuss the idea of a proper subgroup and the trivial subgroup, transitivity of subgroup symbol.
(28:20) Note: Vector spaces are Abelian groups under addition.
(29:07) One-step subgroup test: consider with Z6 under addition mod 6.

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[harshvardhansingh]

Professor, can there be common elements between two subgroups of same order k (where k is prime )of a group apart from the identity element e? If not, how can we prove it ?

[ShwetankT]

Suppose n|(a-b). Hence, a-b = nq --- by division algorithm. Hence, a = nq + b. ie a~b mod n ( by definition). Am i missing something obvious? ie does not a mod n = r automatically means a mod n = r mod n ?

[harshvardhansingh]

Is the proof of the One Step Subgroup test present in the later videos?