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Abstract Algebra

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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 $R$ is nilpotent if $x^n=0$ for some integer $n\ge1$.

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

iii. A homomorphism $A:G\rightarrow G$ of a group $G$ is nilpotent if, 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?

A

If $M\in {\rm UT}_3({\mathbb R})$, then $M$ is a unipotent element of the ring $M_3({\mathbb R})$.

B

${\rm UT}_3({\mathbb R})$ is a nilpotent group.

C

For every $g\in G$, where $G$ is a nilpotent group, the homomorphism $A_g:G\rightarrow G$ given by $h\mapsto [g,h]$ is nilpotent.

D

The group ${\rm T}_3({\mathbb R})$ is nilpotent.

E

The group ${\rm T}_3({\mathbb R})$ is solvable.