First, we need to show that the cosets of N in G are well-defined and partition G.
To show that they are well-defined, we need to show that if a and b are elements of G such that aN = bN, then b^-1a is an element of N.
This can be shown as follows:
aN = bN
=> b^-1aN = N
=> (b^-1a)N = N (using the property that (xN)(yN) = xyN for any x, y in G)
Hence, b^-1a is an element of N.
To show that the cosets partition G, we need to show that every element of G is in exactly one coset of N in G. This follows immediately from the definition of a coset: for any g in G, we have gN = { gn : n in N }. Since every element of G can be written as gn for some g in G and n in N, it follows that every element of G is in exactly one coset of N in G.
Now, we can define the group operation on the cosets of N in G. Let aN, bN be two cosets of N in G. Then we define their product (or multiplication) as follows:
(aN)(bN) = abN
Note that abN is a coset of N in G since N is normal.
We need to show that this operation is well-defined (i.e., it does not depend on the choice of representatives for the cosets) and that it satisfies the group axioms.
To show that the operation is well-defined, we need to show that if aN = a'N and bN = b'N, then abN = a'b'N. This can be shown as follows:
aN = a'N => a' = an for some n in N
bN = b'N => b' = bm for some m in N
=> a'b' = (an)(bm) = ab(nbm) (using the associativity of G)
=> a'b'N = abN (since nbm is in N and thus ab(nbm)N = abN)
Now, we check that the group axioms are satisfied:
- Closure: We just showed that the product of two cosets is a coset.
- Associativity: This follows from the associativity of G.
- Identity: The identity of G/N is the coset containing the identity element e of G, i.e., N itself. It is easy to check that for any aN in G/N, we have (aN)(N) = aN = N(aN), so N acts as the identity element.
- Inverse: The inverse of aN in G/N is the coset containing a^-1, i.e., (aN)^-1 = a^-1N. This is easily verified by checking that (aN)(a^-1N) = N = (a^-1N)(aN).
Hence, we have shown that G/N is a group of order [G:N].
Let N be a normal subgroup of a group G.
Prove that the cosets of N
in G
form a group G/N
of order [G:N]
3 answers
Which is the most likely way in which anti-poverty nongovernmental organizations (NGOs) like Oxfam would help south america to address policital challenges?
Anti-poverty NGOs like Oxfam would most likely help South America address political challenges by advocating for policies and programs that prioritize poverty reduction and social justice, promoting citizen engagement and participation in the political process, and providing support for grassroots social movements that work towards political and economic empowerment for marginalized communities. They may also work on building partnerships between civil society organizations, government agencies, and other stakeholders to promote transparency, accountability, and good governance. Additionally, NGOs may provide capacity-building and training for local organizations and communities to effectively engage in advocacy and address political challenges in their local contexts.
3 answers