Question

सिध्द कीजिये समूह का केंद्र सदैव प्रसामान्य उपसमूह होता है​

Answer 1

SOLUTION

TO PROVE

The centre of a group is a normal subgroup of the group

PROOF

Let G be a group

Then centre Z(G) of G is defined as

Z(G) = { x ∈ G : xg = gx for all g ∈ G }

Now Z(G) is a subgroup of G

We now prove that Z(G) is a normal subgroup of the group G

Let H = Z(G) and let a ∈ G

We have to prove that aH = Ha

Let p ∈ aH

Then p = ah for some h ∈ H

⇒ p = ha since h ∈ Z(G)

So p ∈ aH implies p ∈ Ha

Therefore aH ⊂ Ha

Similarly it can be shown that Ha ⊂ aH

Thus we get aH = Ha for all a ∈ G

∴ H = Z(G) is a normal subgroup of the group G

Hence proved

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