As dtldarek said telecommunications applications offer quite a few such examples. To make this more explicit I mention the following two (that I have tried in the past):

• Diffie-Hellman key exchange. After you have let the students prove that the group $\Bbb{Z}_p^*$ is cyclic (either always or for some specific suitable prime such as $p=107$ or $p=257$), then you can make them wonder, why the problem of solving $m$ from the congruence $$ma\equiv b\pmod{p-1}$$ for a given $a$, coprime to $p-1$ and $b$ is easy, but the "isomorphic" problem of solving $m$ from the congruence $$ g^m\equiv b\pmod p $$ is difficult for a larger $p$ ($g$ a generator of $\Bbb{Z}_p^*$).
• Alamouti coding. This is a method of adding diversity to radio transmission. Transmission from a single antenna will occasionally fade (a moving receiver will occasionally be in a node when a signal combines destructively with a reflected copy of itself). This technique uses two separate antennas transmitting the signal intelligently such that the signals of the two antennas always superpose constructively. It is based on a clever but relatively easy to describe application of the algebra of the division ring of Hamiltonian quaternions.

My students commented on these in their teacher evaluation forms. The response was not unanimously positive (that will never happen), but some did say that such examples convinced them about the usefulness of abstract algebra.

I have mixed feelings about investing a lot of time on a course to examples like this. They do distract the students from the main material a little bit, they are not everybody's cup of tea, and the students may panic into thinking that they need to understand the example fully. The way I did it was to craft a few homework problems to serve as pons asinorum. Then in the homework problem sessions I took the stage (after a student had presented their solution), told them not to take notes, and spent may be 15-20 minutes outlining the application. I have enough control over my problem sessions so that this won't cause the session to run into overtime.

As dtldarek said telecommunications applications offer quite a few such examples. To make this more explicit I mention the following two (that I have tried in the past):

• Diffie-Hellman key exchange. After you have let the students prove that the group $\Bbb{Z}_p^*$ is cyclic (either always or for some specific suitable prime such as $p=107$ or $p=257$), then you can make them wonder, why the problem of solving $m$ from the congruence $$ma\equiv b\pmod{p-1}$$ for a given $a$, coprime to $p-1$ and $b$ is easy, but the "isomorphic" problem of solving $m$ from the congruence $$ g^m\equiv b\pmod p $$ is difficult for a larger $p$ ($g$ a generator of $\Bbb{Z}_p^*$).
• Alamouti coding. This is a method of adding diversity to radio transmission. Transmission from a single antenna will occasionally fade (a moving receiver will occasionally be in a node when a signal combines destructively with a reflected copy of itself). This technique uses two separate antennas transmitting the signal intelligently such that the signals of the two antennas always superpose constructively. It is based on a clever but relatively easy to describe application of the algebra of the division ring of Hamiltonian quaternions.

My students commented on these in their teacher evaluation forms. The response was not unanimously positive (that will never happen), but some did say that such examples convinced them about the usefulness of abstract algebra.

I have mixed feelings about investing a lot of time on a course to examples like this. They do distract the students from the main material a little bit, they are not everybody's cup of tea, and the students may panic into thinking that they need to understand the example fully. The way I did it was to craft a few homework problems to serve as pons asinorum. Then in the homework problem sessions I took the stage (after a student had presented their solution), told them not to take notes, and spent may be 15-20 minutes outlining the application. I have enough control over my problem sessions so that this won't cause the session to run into overtime.

I also use permutation games as examples. 10-15 years ago a significant fraction of my students had played Rotation on their cellphones. Its group theory is very simple. I am retrying it this Spring. I don't expect it to be a hit as the game is probably quite passé now.