How do you show that the centre of a group is a subgroup

To show something is a subgroup we need to show that it satisfies the group axioms. Therefore we need to show that if g and h are in Z(G) then gh is in Z(G), g^-1 is in Z(G), the identity e is in Z(G). As eg = g = ge for all elements g in G we can see e is in Z(G). Then suppose we have g and h in Z(G). Then for all elements j in G we have ghj = gjh as h is in Z(G) = jgh as g is in Z(G). Therefore Z(G) is closed under the group operation. Also we have g^-1 j = g^-1 j e as e is the identity = g^-1 j g g^-1 by definition of inverses = g^-1 g j g^-1 as g is in Z(G) = e j g^-1 = j g^-1 and so g^-1 is in Z(G) and so Z(G) is closed under inverses and is therefore a subgroup of G

AR
Answered by Alex R. Further Mathematics tutor

2915 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

differentiate arsinh(cosx))


Integrate cos(4x)sin(x)


A particle is moving in a straight line with simple harmonic motion. The period of the motion is (3pi/5)seconds and the amplitude is 0.4metres. Calculate the maximum speed of the particle.


If the complex number z = 5 + 4i, work out 1/z.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences