Group algebra ($k[G]$)

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

The matrix-block sizes in the Artin–Wedderburn theorem for $\mathbb{C}[G]$ recover the irreducible character degrees with their multiplicities. They determine the degree-square sum and the number $|G:G^{\prime }|$ of linear characters. Isaacs' character-degree criterion for p-nilpotency therefore detects each normal p-complement and applying it for every prime also detects nilpotent groups. The set of distinct degrees without multiplicities does not contain the same information.

Write the group algebra as $\mathbb{C}[X,X^{-1}]\oplus \mathbb{C}[X,X^{-1}]Y$, with $Y^2=1$ and $YX=X^{-1}Y$. Commuting $f(X)+g(X)Y$ with $X$ forces $g=0$, and commuting with $Y$ forces $f(X)=f(X^{-1})$. Every symmetric Laurent polynomial is a polynomial in $z=X+X^{-1}$, using the recursion $X^{m+1}+X^{-m-1}=z(X^m+X^{-m})-(X^{m-1}+X^{1-m})$. The algebra is free of rank four over this center, with basis $1,X,Y,XY$; the relation $X^2-zX+1=0$ proves spanning, and applying $X\mapsto X^{-1}$ proves independence of $1,X$.

If the order of a finite abelian group $\Delta$ is invertible in the coefficient ring and all its character values belong to that ring, its character idempotents are orthogonal and sum to one. They decompose any module into character components. In a cyclotomic tower with odd $p$, the Teichmüller character provides these components integrally over ${\mathbb{Z}}_p$.

For a conjugacy class $C$ of a finite group, $z_C={\sum }_{g\in C}g$ is central in its group algebra. In a complex irreducible representation of degree $d$ and character of a representation $\chi$, it acts by $|C|\chi (g)/d$; its trace gives this formula and Schur lemma gives scalarity.

The group norm element in the group algebra $kG$ is $N_G={\sum }_{g\in G}g$. It spans the invariant subspace of the left regular group representation, and $N_G^2=|G|N_G$. If the characteristic divides $|G|$, a projection from $kG$ onto $k{N}_G$ would send $N_G$ to both itself and zero, proving that $kG$ is not a semisimple algebra. Its action on indecomposable modules is described by group norm element detects the trivial projective cover.

For a profinite group $G$ and a finite or profinite coefficient ring $R$, the completed group algebra iswhere $U$ ranges over the open normal subgroups. It records convergent noncommutative power series in topological generators of $G$.

Choose ordered minimal topological generators $a_1,\dots ,a_d$ of a uniform pro-p group for odd $p$, put $b_i=a_i-1$, and choose $1/(p-1)<s<1$. Ordered monomials $b^{\alpha }=b_1^{{\alpha }_1}\cdots b_d^{{\alpha }_d}$ give power-series coordinates in the integral completed group algebra. The displayed Non-Archimedean absolute value filtration gives a complete normed ${\mathbb{Q}}_p$-algebra after completion of the rational group algebra. The power relation raises valuation by one, and interchanging $b_i,b_j$ adds terms of valuation at least $1+s>2s$, so multiplication does not decrease total valuation. The p-adic logarithm and p-adic exponential are inverse on the relevant small balls. In particular $\log G$ is a faithful realization of the Lie algebra of a uniform pro-p group, useful for verifying its intrinsic limit operations and the Baker--Campbell--Hausdorff formula without assuming the correspondence in advance.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A p-adic measure gives the compatible finite-quotient elements ${\sum }_{gU}\mu (gU)[gU]\in I[G/U]$. The transition maps sum coefficients over fibers. Conversely compatible coefficients define values on cosets, then all clopen sets, and hence integration. Thus the inverse limit Iwasawa algebra is the ring of measures, with multiplication given by convolution of p-adic measures. Completeness of $I$ suffices even when its residue field is infinite.

Choose a topological generator $\gamma$ of the Galois group of a Zp-extension and set $T=\gamma -1$. The completed group algebra becomes the formal power series ring ${\mathbb{Z}}_p[[T]]$, a two-dimensional complete regular local integral domain with maximal ideal $(p,T)$. Its finite-layer quotient is $\Lambda /((1+T{)}^{p^n}-1)\cong {\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$.

A homomorphism between finitely generated one-variable Iwasawa modules is a pseudo-isomorphism if its kernel and cokernel are finite. Finite errors do not change the Iwasawa-module rank or the characteristic ideal of a torsion module. They can still change its integral structure, so a pseudo-isomorphism is weaker than an isomorphism.

For a finitely generated module over the integral domain $\Lambda$, its Iwasawa-module rank counts free summands up to pseudo-isomorphism. Rank zero is equivalent to being a torsion module. A positive rank $r$ forces the ${\mathbb{Z}}_p$-ranks of finite-layer coinvariant modules to grow like $r{p}^n+O(1)$.

A filtration on a group $G$ is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is separated when $\omega (x)=\infty$ only for $x=1$.

For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a p-valued group.

An ordered basis $(g_1,\dots ,g_d)$ of a complete finite-rank p-valued group gives each element a unique convergent expression $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i\in {\mathbb{Z}}_p$, and its valuation is the minimum of $\omega (g_i)+v_p({\lambda }_i)$ over the nonzero coordinates.

A p-valued group is p-saturated when it is complete and every $x$ withhas a pth root in the group.

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.

For a p-valued group $(G,\omega )$, give $p^r(g_1-1)\cdots (g_s-1)$ weight $r+{\sum }_i\omega (g_i)$. The spans of elements of weight at least $\lambda$ form a multiplicative filtration on ${\mathbb{Z}}_p[G]$ and its completions.

The initial form of $g-1$ depends only on the initial form of $g$ in $\mathrm{gr}G$. The resulting Lie map extends to a surjective graded algebra homomorphismFor a complete finite-rank p-valued group, ordered-basis expansions show that it is an isomorphism.

If $G$ is a complete finite-rank p-valued group with center $Z(G)$, thenThe key facts are that the finite-conjugacy center of a p-valued group equals its center and that compatibility through the finite group-algebra quotients eliminates coefficients on infinite conjugacy classes.

Inside the completed rational Iwasawa algebra, the convergent seriessends a finite-rank p-saturated group to a ${\mathbb{Z}}_p$-Lie algebra. The Baker--Campbell--Hausdorff formula and its commutator expansion prove closure under addition and the commutator bracket.

If $(g_1,\dots ,g_d)$ is an ordered basis of a finite-rank p-saturated group, thenwhere a primitive element $x$ satisfies $\Delta (x)=x\otimes 1+1\otimes x$ for the completed Hopf algebra coproduct.