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Let

(G,\cdot)

be a finite group with the identity element,

e

. The smallest positive integer

n

with the property that

x^{n}= e

, for all

x \in G

, is called the exponent of

G

. (a) For all primes

p \geq 3

, prove that the multiplicative group

\mathcal G_{p}

of the matrices of the form

\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}

, with \hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z, is not commutative and has exponent

p

. (b) Prove that if

\left( G, \circ \right)

and

\left( H, \bullet \right)

are finite groups of exponents

m

and

n

, respectively, then the group

\left( G \times H, \odot \right)

with the operation given by

(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)

, for all

\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H

, has the exponent equal to

\textrm{lcm}(m,n)

. (c) Prove that any

n \geq 3

is the exponent of a finite, non-commutative group. Ion Savu

Problem Statement