1. ## 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 $\displaystyle G = {Z_{21}}^{x}$, show that $\displaystyle H = {[x]_{21}| x\equiv 1 (mod 3)}$ and $\displaystyle K = {[x]_{21}| x\equiv 1 (mod 7)}$
are subgroups of G.

2. ## Re: group theory

first of all $\displaystyle (\mathbb{Z}_{21})^\times$ is the group of units of the integers modulo 21 (all elements of $\displaystyle \mathbb{Z}_{21}$ 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 $\displaystyle (\mathbb{Z}_{21})^\times = \{1,2,4,5,8,10,11,13,16,17,19,20\}$

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 $\displaystyle (\mathbb{Z}_{21})^\times$, 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!

3. ## 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!