What condition must a group homomorphism \(\phi:G\to K\) satisfy?

What is the kernel of a homomorphism \(\phi:G\to K\)?

What is the image of a homomorphism \(\phi:G\to K\)?

If \(\phi\) is a homomorphism, where must it send the source identity \(e_G\)?

Let \(\phi:\mathbb Z\to\mathbb Z_3\) reduce integers modulo \(3\). Compute \(\phi(8)\).

For \(\phi:\mathbb Z\to\mathbb Z_3\) reducing modulo \(3\), compute \(\phi(4+5)\) and \(\phi(4)+\phi(5)\).

For the modulo \(3\) map \(\phi:\mathbb Z\to\mathbb Z_3\), list three elements of \(\ker\phi\).

Define \(\phi:\mathbb Z\to\mathbb Z\) by \(\phi(n)=2n\). Is \(\phi\) a homomorphism under addition?

For \(\phi:\mathbb Z\to\mathbb Z\) with \(\phi(n)=2n\), find the kernel and image.

Is \(f:\mathbb Z\to\mathbb Z\), \(f(n)=n+1\), a homomorphism under addition?

Show that a homomorphism sends inverses to inverses.

Why must the source and target operations both be used when checking a homomorphism?

Let \(\phi:\mathbb R\to\mathbb R_{>0}\) be \(\phi(x)=e^x\), with addition in the source and multiplication in the target. Prove it is a homomorphism.

For \(\phi(x)=e^x\) from \((\mathbb R,+)\) to \((\mathbb R_{>0},\cdot)\), find the kernel.

Let \(\det:GL_2(\mathbb R)\to\mathbb R^*\) send a matrix to its determinant. Explain why this is a homomorphism.

For the determinant homomorphism \(GL_2(\mathbb R)\to\mathbb R^*\), describe the kernel.

If \(\phi:G\to K\) is a homomorphism and \(\ker\phi=\{e_G\}\), explain what this says about elements with the same image.

A student says any function between groups is a group map because it sends elements to elements. Diagnose the error.

Prove that the kernel of a homomorphism is a subgroup of the source group.

Prove that the image of a homomorphism is a subgroup of the target group.