In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then gH = {gh : h an element of H?} is a left coset of H in G, and Hg = {hg : h an element of H?} is a right coset of H in G. Only when H is normal will the right and left cosets of H coincide, which is one definition of normality of a subgroup. A coset is a left or right coset of some subgroup in G. Since Hg = g?(?g-1Hg?), the right cosets Hg (of H?) and the left cosets g?(?g- 1Hg?) (of the conjugate subgroup g-1Hg?) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left cosets and right cosets are the same then H is a normal subgroup and the cosets form a group called the quotient group. The map gH?(gH)-1=Hg-1 defines a bijection between the left cosets and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the index of H in G. For abelian groups, left cosets and right cosets are always the same. If the group operation is written additively then the notation used changes to g+H or H+g. Cosets are a basic tool in the study of groups; for example they play a central role in Lagrange\'s theorem. Examples et G be the multiplicative group of {-1,1}, and H the trivial subgroup (1,*). Then -1H={-1}, 1H=H are the sole cosets of H in G. Let G be the additive group of integers Z = {..., -2, -1, 0, 1, 2, ...} and H the subgroup mZ = {..., -2m, -m, 0, m, 2m, ...} where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ+1, ... mZ+(m-1), where mZ+a={..., -2m+a, -m+a, a, m+a, 2m+a, ...}. There are no more than m cosets, because mZ+m=m(Z+1)=mZ. The coset mZ+a is the congruence class of a modulo m.[1] Another example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector space form an abelian group under vector addition. It is not hard to show that subspaces of a vector space are subgroups of this group. For a vector space V, a subspace W, and a fixed vector a in V, the sets are called affine subspaces, and are cosets (both left and right, since the group is abelian). In terms of geometric vectors, these affine subspaces are all the \"lines\" or \"planes\" parallel to the subspace, which is a line or plane going through the origin Solution In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then gH = {gh : h an element of H?} is a left coset of H in G, and Hg = {hg : h an element of H?} is a right coset of H in G. Only when H is normal will the right and left cosets of H coincide, which is one definition of normality of a subgroup. A coset is a left or right coset of some subgroup in G. Since Hg = g?(?g-1Hg?), the right cosets Hg (of H?) and the left cosets g?(?g- 1Hg?) (of the conjugate subgroup g-1Hg?) are the same. Hence it is .

1. In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G,
then gH = {gh : h an element of H?} is a left coset of H in G, and Hg = {hg : h an element of H?}
is a right coset of H in G. Only when H is normal will the right and left cosets of H coincide,
which is one definition of normality of a subgroup. A coset is a left or right coset of some
subgroup in G. Since Hg = g?(?g-1Hg?), the right cosets Hg (of H?) and the left cosets g?(?g-
1Hg?) (of the conjugate subgroup g-1Hg?) are the same. Hence it is not meaningful to speak of a
coset as being left or right unless one first specifies the underlying subgroup. In other words: a
right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left
cosets and right cosets are the same then H is a normal subgroup and the cosets form a group
called the quotient group. The map gH?(gH)-1=Hg-1 defines a bijection between the left cosets
and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The
common value is called the index of H in G. For abelian groups, left cosets and right cosets are
always the same. If the group operation is written additively then the notation used changes to
g+H or H+g. Cosets are a basic tool in the study of groups; for example they play a central role
in Lagrange's theorem. Examples et G be the multiplicative group of {-1,1}, and H the trivial
subgroup (1,*). Then -1H={-1}, 1H=H are the sole cosets of H in G. Let G be the additive group
of integers Z = {..., -2, -1, 0, 1, 2, ...} and H the subgroup mZ = {..., -2m, -m, 0, m, 2m, ...}
where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ+1, ... mZ+(m-1),
where mZ+a={..., -2m+a, -m+a, a, m+a, 2m+a, ...}. There are no more than m cosets, because
mZ+m=m(Z+1)=mZ. The coset mZ+a is the congruence class of a modulo m.[1] Another
example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector
space form an abelian group under vector addition. It is not hard to show that subspaces of a
vector space are subgroups of this group. For a vector space V, a subspace W, and a fixed vector
a in V, the sets are called affine subspaces, and are cosets (both left and right, since the group is
abelian). In terms of geometric vectors, these affine subspaces are all the "lines" or "planes"
parallel to the subspace, which is a line or plane going through the origin
Solution
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G,
then gH = {gh : h an element of H?} is a left coset of H in G, and Hg = {hg : h an element of H?}
is a right coset of H in G. Only when H is normal will the right and left cosets of H coincide,
which is one definition of normality of a subgroup. A coset is a left or right coset of some
subgroup in G. Since Hg = g?(?g-1Hg?), the right cosets Hg (of H?) and the left cosets g?(?g-
1Hg?) (of the conjugate subgroup g-1Hg?) are the same. Hence it is not meaningful to speak of a
coset as being left or right unless one first specifies the underlying subgroup. In other words: a
right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left

2. cosets and right cosets are the same then H is a normal subgroup and the cosets form a group
called the quotient group. The map gH?(gH)-1=Hg-1 defines a bijection between the left cosets
and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The
common value is called the index of H in G. For abelian groups, left cosets and right cosets are
always the same. If the group operation is written additively then the notation used changes to
g+H or H+g. Cosets are a basic tool in the study of groups; for example they play a central role
in Lagrange's theorem. Examples et G be the multiplicative group of {-1,1}, and H the trivial
subgroup (1,*). Then -1H={-1}, 1H=H are the sole cosets of H in G. Let G be the additive group
of integers Z = {..., -2, -1, 0, 1, 2, ...} and H the subgroup mZ = {..., -2m, -m, 0, m, 2m, ...}
where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ+1, ... mZ+(m-1),
where mZ+a={..., -2m+a, -m+a, a, m+a, 2m+a, ...}. There are no more than m cosets, because
mZ+m=m(Z+1)=mZ. The coset mZ+a is the congruence class of a modulo m.[1] Another
example of a coset comes from the theory of vector spaces. The elements (vectors) of a vector
space form an abelian group under vector addition. It is not hard to show that subspaces of a
vector space are subgroups of this group. For a vector space V, a subspace W, and a fixed vector
a in V, the sets are called affine subspaces, and are cosets (both left and right, since the group is
abelian). In terms of geometric vectors, these affine subspaces are all the "lines" or "planes"
parallel to the subspace, which is a line or plane going through the origin