Moderate# Groups and Rings: When "Potent" is Part of a Definition

ABSALG-LL1ECB

Let $G$ be a group with neutral element $e$. A normal series (that is $G_i\trianglelefteq G$, for all $i$):

$$ \{e\}=G_0\trianglelefteq G_1\trianglelefteq\ldots\trianglelefteq G_n=G $$

...is called *central* if, for all $i=0,\ldots,n-1$, we have $G_{i+1}/G_i\leq C(G/G_i)$, where $C(G/G_i)$ is the center of $G/G_i$.

Equivalently, we have $[G_{i+1},G]\leq G_i$ for all $i=0,\ldots,n-1$, where $[G_{i+1},G]$ is the group generated by the commutators $[g_{i+1},g]$, where $g_{i+1}\in G_{i+1}$ and $g\in G$.

A group $G$ is called *nilpotent* if it has a central series of finite length $n$ and the length of its smallest central series is called its *nilpotency class*, that of the trivial group $\{e\}$ being zero.

Consider also the following occurrences of *potent* in mathematical definitions:

i.An element $x\in R$ of a ring $R$ isnilpotentif $x^n=0$ for some integer $n\ge1$.

ii.An element $u\in R$ of a ring $R$ isunipotentif $u-1$ is a nilpotent.

iii.A homomorphism $A:G\rightarrow G$ of a group $G$ isnilpotentif, for some integer $n\ge 1$, we have $A^n(g)=e$, for all $g\in G$.

**Let:** $M_3({\mathbb R})$ be the ring of $3\times 3$ matrices with entries in ${\mathbb R}$.

**Let:** ${\rm UT}_3({\mathbb R})$ be the group of $3\times $3 uni-triangular matrices with entries in ${\mathbb R}$: they are the upper triangular matrices with all diagonal entries equal $1$.

Let ${\rm T}_3({\mathbb R})$ be the group of upper triangular matrices with non-zero determinant.

Which of the following is **FALSE**?