
group theory
I am confused by the notation in this problem, i have an idea about showing a group to be a subgroup, but i am not sure what the notation means and i am not seeing it anywhere in my books or online, i am hoping someone can help.
In , show that and
are subgroups of G.

Re: group theory
first of all is the group of units of the integers modulo 21 (all elements of that have a multiplicative inverse with respect to multiplcation modulo 21).
it turns out, that these are precisely the integers between 1 and 20 that are relatively prime to 21 (or more precisely, the equivalence classes of these integers mod 21).
so
which of these numbers are congruent to 1 mod 3?
clearly that set is {1,4,10,13,16,19} (note that this set has order 6, which divides 12).
which of these numbers are congruent to 1 mod 7?
these: {1,8}. note that this set also has order a divisor of 12.
i'll show that {1,8} is a cyclic group of order 2, under multiplication mod 21. to do this, it suffices to show that 8 is of order 2:
8^2 = 8*8 = 64 = 1 (mod 21) (since 64 = 63 + 1 = 21(3) + 1).
if {1,4,10,13,16,19} is indeed a subgroup of , then it is abelian (since multiplication mod 21 is commutatitive), and as such it must be cyclic of order 6. find a generator, and you're done!

Re: group theory
Thank you so much for this, i was really unsure about the meaning of the notations. this helps a whole lot. because what i have seen in the texts thus far could have well been Greek!