Norm (abelian group)

In the context of abelian groups, the term "norm" typically refers to a homomorphism between groups, particularly used when studying Galois extensions and group cohomology. The norm map provides a way to "average" elements over the action of a group. More formally:

Let $G$ be a finite group and let $A$ be a $G$-module, which means that $A$ is an abelian group and there is a group action of $G$ on $A$, denoted by $g \cdot a$ for $g \in G$ and $a \in A$, such that $g \cdot (a + b) = g \cdot a + g \cdot b$ and $(g_1g_2) \cdot a = g_1 \cdot (g_2 \cdot a)$ for all $g, g_1, g_2 \in G$ and $a, b \in A$.

The norm map (or trace map) $N: A \rightarrow A$ is defined by

$N(a) = \sum_{g \in G} g \cdot a$

where the sum is taken in the abelian group $A$.

Equivalently, if we index the elements of $G$ as $g_1, g_2, \dots, g_n$, then

$N(a) = g_1 \cdot a + g_2 \cdot a + \dots + g_n \cdot a$.

The image of this map, $N(A)$, is a subgroup of $A$. The kernel of this map is often of interest, especially in connection with Tate cohomology.

The norm map is a homomorphism because for any $a, b \in A$,

$N(a+b) = \sum_{g \in G} g \cdot (a+b) = \sum_{g \in G} (g \cdot a + g \cdot b) = \sum_{g \in G} g \cdot a + \sum_{g \in G} g \cdot b = N(a) + N(b)$.

Furthermore, the norm map is $G$-equivariant in the sense that if $h \in G$, then

$h \cdot N(a) = h \cdot \sum_{g \in G} g \cdot a = \sum_{g \in G} h \cdot (g \cdot a) = \sum_{g \in G} (hg) \cdot a$.

Since $G$ is a group, the set ${hg \mid g \in G}$ is equal to $G$, so

$\sum_{g \in G} (hg) \cdot a = \sum_{g \in G} g \cdot a = N(a)$.

Therefore, $h \cdot N(a) = N(a)$ for all $h \in G$. This means that the image of the norm map, $N(A)$, is contained in the subgroup of $A$ consisting of elements fixed by the action of $G$, denoted $A^G = {a \in A \mid g \cdot a = a \text{ for all } g \in G}$. Thus, we often consider the norm map as $N: A \to A^G$.

In particular, when $A$ is the multiplicative group of a Galois extension $L/K$ with Galois group $G = \text{Gal}(L/K)$, the norm map takes an element $\alpha \in L^\times$ and maps it to the product of all its Galois conjugates:

$N_{L/K}(\alpha) = \prod_{\sigma \in G} \sigma(\alpha)$.

This norm is an element of $K^\times$. The terminology "norm" in this setting stems from algebraic number theory and field theory. The relationship between the group cohomological definition and this Galois extension definition can be made precise by letting $G$ act on $L^\times$.

The norm map is a crucial tool in the study of group cohomology, Galois cohomology, and class field theory, connecting the arithmetic of fields with the structure of their Galois groups.