MATH1061/7861, Thu 8 Oct 2020
Question 1. From yesterday: Prove that the group (Q − {0}, ×) is not cyclic.
DoesCQ903 e s
ie every qtQEo3
f
let p prime not in prime dec of sis mum
Repeated group ops yx2 Idem
Then p f si fer auf n t I.LI
tsM behave same
st 22
I
so L
c
Two
groups
ta
farFonenE7 L
for some e
s
23
t on.sn
of 6
Equivalence
knots
surfaces
ilazer groups pg
theorem
moth I
t3 3
Question 2. Which pairs of groups are isomorphic, and why / why not? ◦ (Z12, +) and (Z2×Z2×Z3, +)
99
◦ (Z60, +) and (Z2×Z3×Z10, +) u
sit.EE
If
to iso fl3fHid idM m
Tmod 3
f3 tf3etc3a if
j fad
id
33
id p
IE Testino
are G H isomorphic Impossibles
3g
I sieie I
Iaica.I.ied
f3tf3 f33fnotie notid
fC3 tfc3 effs fC3ez f not id i not 1d
knots
F
LHS RHS
mod 3 since 360
8
4 1 4 Gay
◦ (Z60, +) and (Z3×Z4×Z5, +) YES f n mod 60
Ln
mod 4 0 32
n mod 5
eg
ft27 jNudft
mod 3 n fo0oo
ii mod ymodt gonads
g
c mod 3 tcgmodDmod3
mods 5 60
x wed4 t zoned4 Mod 4 5
g mod3 Gaymad4 scey meet 5
Ln mod 3 Oh Bijection
f
nggmggod6o
nmod4 nmod5 yez
6
o O
FE.io
57
o BIJECTION
GD
◦ (Z, +) and (Z×Z, +)
NI 1 47 cyclic
It 7L not cyclic
GD
ever get
◦ (Z6, +) and (Z7 − {0}, ×)
EtyeRes4sEi only leos Not
InCR63e
ka ks both Con and lo
◦ (R, +) and (R − {0}, ×)
◦ (R, +) and (R+, ×)
IT
Ct f 1 tD D
fGuy Its
◦ (Q, +) and (Q+, ×)
◦ (Z2×Z, +) and (Z3×Z, +)
3 sE
Is floc fCg ISOMORPHKI.tl
For
a
n
f
Jf GRDNes sitIf O
2
0 Ex
Monday
2750341315 Know d
Challenge:
◦ Is (Q, +) isomorphic to (Q × Q, +)? ◦ Is (R, +) isomorphic to (R × R, +)?
Question 3. Prove that, for all d,n in Z+, if d |n then (Z_n, +) has a subgroup isomorphic to (Z_d, +).