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If youhave agroup
any
subset will have associativeproperty but
closure
property is not associative
Theorem let GA be agroup and H be asubset of G Then
CH is the followinghold
a subgroup
ofG iff
F quest His nonempty Closure
f
my EH
K EH
quantifier
a
x
YEH
EM Yo prove handanot immerse
taha huncha
Proof
By A is nonempty and the 1st and4thproperties
assumption
of a group hold However since is associative
for G it is also
associative
of any subset ofG
Show H contains an identity
ut e be the identityforG Since His non empty Fat H
se E H x a e e H Thus H contains the identity
By By
Therefore C D is a subgroup of CG
Toshow asubset
if
is asubgroup 3 ta property
hold huna parcha
Proving and now
O
T hungryo Usingmatrices Ga Ma RD t
let te
E a b c ER
Prove H t is a subgroup
, I e D gray
proof l Euta example diney F qu EH Thus H
Iggy is non empty
Proof ly g
let E H Then
8 2 an to
CA uz E EH
Thus Closure properly holds
prooff let a
E EH Then E Tete
Thus the inverseproperty holds
Thus H t is a subgroup holds
Ex Operating on G Ma R t
b c de R at'd te
ut k
I k a
cnn.ae
I 9 Ek since 2 51 7 1 0
Proof let
ad I E
e k Then at sd 70 0
and w
5774 0
so land
I Jtf can
KY Cdt
Ek since
Then 5 dtz 7 ay
fatw
, a I C I
at w t 5 d 52 7C 74
wt 52 74 sd 7C
0 0
property hold
Thus the closure
I I g
Proof ut e k Then e k
q
since t a Std 7 e
a sd 7C
I a sd 70 1 0 0
Therefore the inverse property holds
Therefore K is a subgroup
is a set with a
binaryoperation of
multiplication and an identity
At set
of elements in A with a multiplicativeinverse
En R R o
Ma R is the set of 2 2 invertible matrices i.e
determinant is non zero
Eba
at Lad be determinant o bhayo bhaney invertible
hahdaina so inverse hahdaina
qÉÉÉ Éqb a b c de R ad be I
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