Prove a mapping is a normal subgroup

Given a problem like this... Let @:G-->H be a homomorphism of G onto H, and let N be a normal subgroup of G. Show that @(N) is a normal subgroup of H. How do I prove that a mapping is a normal subgroup of a group? What I am missing here is some understanding of the terminology and some clear understanding of mappin ...continues

Subgroups and indexes

Explain what the index of a subgroup and a coset of a group are. Also, prove that if N is a subgroup of a group G such that [G: N] = 2, and if "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.

Conjugacy of group elements

Show that conjugacy of group elements is an equivalence relation.

Conjugacy Classes

What are the conjugacy classes in S_3?

Equivalence relations, surjective maps, partitions and fibers??

Consider any surjective map f from a set X onto another set Y. We can define a relation on X by x_1 ~ x_2 if f(x_1) = f(x_2). Check that this is an equivalence relation. Show that the associated partition of X is the partition into "fibers" f^(-1) (y) for y in Y. I would like to understand what this question is asking me a ...continues

RINGS - prove:(1)multiplicative identity is unique, (2)left & right multiplicative inverses are =

I need to understand how to show/prove the following regarding a ring R: 1) if a ring R has a multiplicative identity, then the multiplicative identity is unique. 2) if an element r that is in the ring R has a left multiplicative inverse r' and a right multiplicative inverse r", then r' = r".

Show that a set of matrices is a ring without identity element

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n. I need to understand how to show that this set of matrices is a ring w ...continues

Show that a ring isomorphism has a multiplicative identity and is commutative

Suppose φ:R --> S is a ring isomorphism. Show that R has a multiplicative identity if, and only if, S has a multiplicative identity. Show that R is commutative if, and only if, S is commutative.

Show that the set of real rational functions is a field.

NOTE: In this description, R represents the symbol for the set of real numbers. I couldn't find a way to type or copy the correct R symbol for the set of real numbers. Also, the parentheses in R(x) is used to distinguish the ring R(x) of rational functions from the ring R[x] of polynomials. Show that the set R(x) of rational ...continues

Rings: left and right ideals

Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x. Similarly, xR = {xr: r in R} is the principal right ideal generated by x.