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Group theory is like a nail. By themselves, you can’t build very much out of nails alone. And yet if you are building anything, they are probably in there somewhere. Just so, groups are fundamental building blocks, and appear everywhere. However, you usually need a little bit more than just group theory itself to make full use of it. That said, here is a short list of various places where groups show up.Galois groups are fundamental objects of Galois theory. Galois theory can be sort of summed up (although it is an oversimplification) as the study of how roots of polynomials can be shuffled around while maintaining algebraic relationships. Important results of Galois theory include:A nice proof that the quintic and other higher order polynomials do not in general have roots that can be expressed in terms of radicals (I wrote about this here).Using similar ideas, you can prove various impossibility results in geometry, such as the fact that you cannot trisect an angle using a straightedge and compass.Differential Galois theory is analogous to ordinary Galois theory, but studies differential operators. Groups are also fundamental here, although they tend to be matrix Lie groups, rather than finite groups. Differential Galois theory allows one to study how and when some functions are derivatives of other functions. This has important consequences:Liouville’s theorem is a result that allows you to prove that various functions like [math]e^{x^2}[/math] and [math]\sin(x)/x[/math] do not have elementary anti-derivatives (i.e. they cannot be written as compositions of simple functions like polynomials, [math]1/x[/math], trigonometric functions, exponential functions, and logarithms). (Although as Ryan Reich points out in the comments, the proof of Liouville’s theorem actually uses almost no group theory, so this including this is a bit cheat-y.)If I’m not mistaken, the Risch algorithm is built up using ideas of differential Galois theory. The Risch algorithm is a marvelous semi-algorithm that takes in elementary functions and tells you: 1) whether they have elementary anti-derivatives, and 2) if they do, gives you what they are. In short, it is what allows computer algebra systems to automatically compute integrals.Lie groups are groups that are “smooth”—i.e. groups that you can also think of as surfaces. These show up in the study of partial differential equations; this is because if you can show that solutions to your partial differential equations satisfy some type of symmetry, this can significantly decrease the difficulty of solving the equation. Of course, Lie groups precisely capture such symmetries. This is such a fundamental relationship that it has become a cornerstone of modern quantum field theory.Groups allow you to study and quantify the symmetries of a space. This has applications, from the silly to the deeply useful.All of the different ways that you can move a Rubik’s cube form a group. If you understand the generators and the relations between them, you can build algorithms allowing you to solve a Rubik’s cube.In chemistry, the symmetries of crystal lattices are studied and described in terms of group theory.You can study geometry from a group theoretic perspective, by considering the space of transformations that preserve properties that we are interested in—or, conversely, figuring out what properties are preserved if we choose a particular space of transformations. For instance, there are problems we can solve via inversive geometry, where we consider linear fraction transformations. (The first part of this answer basically did exactly that.) It is very common to study hyperbolic space this way, too, because usually the distance functions are messy to deal with directly.The study of finite abelian groups develops alongside elementary number theory, and there are many results in one that prove results in the other (and vice versa). In particular, Euler's theorem is most naturally proved as a group theoretic result. But, of course, Euler’s theorem, together with the (group theoretic) observation that every group [math]\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times[/math] is cyclic (if [math]p[/math] is an odd prime), is a fundamental building block in the Diffie–Hellman key exchange, and many other cryptographic protocols are likewise built on fundamental group theory/elementary number theory.The group structure of elliptic curves is of primary importance in studying them. Elliptic curves themselves are incredibly useful, and have myriad applications.Elliptic curve cryptography is very nice, because it can give the same level of security as more classic algorithms, but with much shorter keys (which are easier to keep secret).There are various Diophantine equations that can be solved by appealing to the properties of elliptic curves (e.g. see Alon Amit's answer to How do you find the positive integer solutions to [math]\frac{x}{y+z}+ \frac{y}{z+x}+\frac{z}{x+y} = 4?[/math], or Senia Sheydvasser's answer to What are all solutions to the Diophantine equation, [math]x^3-2y^2=-1[/math]?)Elliptic curves were central to the proof of Fermat’s last theorem.Fundamental groups are basic objects in algebraic topology, as are homology and cohomology groups, for that matter. All of these different objects let you prove all sorts of nice topological things, and allow you to show that various surfaces and spaces are different topologically (e.g. you cannot deform one into the other). See, for example, Homology and Dimensions.This is barely scratching the surface, but it should be a good first introduction.

Conservatives tend to believe in individual liberty, limited government and the free market system. Unlike progressives that seem to exist in a world where feelings are all that matter, Conservatives also seem to have a better understanding of history and human experience, so they know that living wage systems are ultimately highly abusive to the very people we want to help.The cornerstone of the free market system is a vast and dispersed marketplace where parties engage in voluntary transactions based on their own notion of enlightened self-interest. Let’s say you shop for a phone and choose the latest iPhone…you get a product you want, and Apple gets some of your money - it’s a voluntary and mutually beneficial transaction.Same is true for labor. Whether I’m a genius economist working for Goldman-Sachs or a low-skilled laborer that can barely speak English, I want to be able to “sell” my labor to someone and be paid fairly for it.What determines the value of my labor? Well, people that believe in liberty and free markets tend to believe labor should be priced according to its value. This is ultimately three things: productivity, supply and demand, and cost of alternatives.On the other hand, people believing in things like a “living wage” disconnect the value of labor from wages. If Sally needs $80K/year to live, then that’s what her wages should be - even if the value of what she produces is much less than this. If you think about it, this is pure Marxism: “from each according to their abilities, to each according to their needs”.Ask yourself, how likely are you to pay two or three times the value for any product and you understand the dilemma facing Sally…government may put a price tag on her labor, but if it’s not in line with value, she’s going to sit unused because nobody will buy her.The other important consideration is the relationship of government to the free market.Conservatives believe in the voluntary and open nature of the free market. If I want to enter into a contract to sell my labor to someone, who is the government to substitute its notion of fairness for that of the parties involved?Let’s say I want to hire my teenage neighbor to babysit for a few hours a week, and it’s worth $5/hour to me. My neighbor could use the extra cash, and I can use the help, so this is a fair and mutually beneficial transaction. Except that the government technically doesn’t let me pay her $5/hour…it demands I pay her $15/hour. At this price, I might as well pay to put my kid in some sort of preschool program, and so I don’t hire my neighbor. I pay more than I want, my neighbor goes unemployed.To say it more harshly, by substituting it’s preferences for that of the parties, the government demands that my teenage neighbor be unemployed rather than earning what her work is actually worth in the free market. Government trampled her right to earn a wage, my right to hire her - and in so doing, everyone is worse off. Conservatives thus reject government intervention in private transactions as an attack on the individual liberties of us all.The final thing is that Conservatives know based on history and experience that when you tamper with free markets, you tend to get bad outcomes.As one of the other answers to this question mentions, only 3% of hourly workers earn the minimum wage today. Not mentioned is that in 1970, this number was 9%…somehow, we’ve managed to destroy two-thirds of the market for minimum wage jobs.The part lost on progressives is that government doesn’t have some magical power to suspend the laws of economics and human nature. If there’s a disparity between the price of labor and the value of labor, businesses will find different solutions, and while this might be good for business, it’s usually not good for labor.An example is manufacturing. In 1970, about a third of all US minimum wage jobs were in the manufacturing sector…today, almost zero of these jobs exist. Why? No company can exist paying workers substantially more than the value of their labor. When the cost of labor crosses a certain threshold, it becomes more beneficial to offshore that job out of the country, or to replace the worker with automation - and this is exactly what happened.Feel-good politics and progressivism therefore create more problems and pain for the people they want to help, as evidenced by the shrinkage in low-skilled labor. Look at unemployment rates in the 18–24 age group, especially in the inner cities and among minority groups. It’s a national disgrace - and much is the fault of all the various entities trying to force “living wages” on an economy. It might let progressives virtue signal and inflate their fragile egos, but this comes at the price of millions trapped in poverty.So there you have it.“Living wages” are an unnatural attempt by progressives to tamper with the laws of nature, and Conservatives, understanding free markets and human nature are therefore much more likely to do the truly compassionate thing. We know that by getting government out of the way and letting capitalism work its miracles, people will ultimately be much better off than any Marxist-progressive policy can ever be.

[Updated: 14/03/21- I apologise in advance for the length!]A few characteristics I’ve identified among those I consider highly intelligent:Qualifying language: When an intelligent individual makes a statement, it is unlikely to be expressed definitively. An example might be, rather than using ‘I will have this project finished by the end of the day’, they might say, ‘I will most likely have the project finished by the end of the day.’ This ‘hedges’ the speaker against the negative outcome of not completing the project on time; less intelligent individuals making promises of completion don’t think ahead to guard against the potential damage to their credibility should they not complete the project.Listening: Within the dynamic of conversation, intelligence is often expressed in active listening rather than sharing of information. Intelligence is highly correlated with capacity for introspection. The intelligent individual leans toward gathering as much data as possible in order to come to a fuller understanding of their conversation partner’s position, as well as to offer more coherent arguments themselves. Internalising the content of the conversation, this individual is likely to only speak when expected to, or if they have a specific insight to offer.‘The Grey Eminence’: This intellectual archetype is associated with the intelligent individual’s propensity to listen, rather than engage directly. When a ‘high IQ’ person seeks to influence society, they often employ ‘front men’, such as publicly known politicians, in order to present policies and ideas. This is another method of ‘hedging’ and reducing risk to the intelligent individual. Rather than presenting their (possibly controversial) argument directly, they assume the role of the ‘Grey Eminence’; a clandestine and influential figure behind the curtain, and unknown to the wider population. They are thus able to direct events without assuming responsibility, accountability or danger. This role does not only present in politics, but will also exist in more intimate situations, such as family units or small organisations.Recognised Competence: In almost every organisation, there will be at least one ‘go-to guy’ that gets more than their fair share of requests for help and technical assistance. It may seem unfair to disproportionately burden one person with the problems of others, but this phenomenon is simply a recognition of the asymmetry of competence (see the Pareto 80/20 rule); there will always be employees that are more competent than others, due either to experience or talent. When a colleague is recognised as highly competent, this can be used as a relatively accurate proxy for above-average IQ.Efficiency: This term should be understood in a general sense. Many intelligent individuals through history have in fact been seen as ‘lazy’ by their contemporaries. However, intelligence allows for pre-planning and directing activity in an efficient manner. An emphasis on efficiency accomplishes the goal of the activity in a way that doesn’t waste the time or resources of the intelligent person. This also commonly manifests in the communication style of the intelligent person; small talk or conversation about ‘superficial’ topics is avoided. Instead, they will try to guide the conversation on to either the information they are trying to extract, or to more interesting, conceptual, and ‘big picture’ topics.Messy: This is a subsidiary characteristic of efficiency. While to the majority of the population cleaning and ordering one’s room seems an essential task, the intelligent person may view this as tedious and unnecessary ‘busy work’ that takes them away from their intense contempletion of more important subjects and ideas. Family members may express apprehension or disgust at the state of the intelligent person’s living situation, to which the intelligent person will again be quite dismissive or oblivious. This ‘head in the clouds’ trait will sometimes result in poor personal hygiene that hampers their social prospects.Humility: The philosopher Socrates once said, ‘I know that I know nothing’. This is the cornerstone to humility, and a necessary characteristic of intelligent, as well as wise people. I would perhaps argue that it is impossible to be truly intelligent without a complementary degree of wisdom. The intelligent individual will not land on an answer or solution without some second-guessing of their assumptions. This ‘checking’ instinct is imperative in how intelligent people refine raw data down to come to more accurate conclusions.Anti-Utopian: As natural skeptics, intelligent people tend toward pragmatism, and are wary of ‘better world’ narratives. This is not to say that the intelligent cannot envision alternative futures they would personally prefer, but if presented with the argument that a certain solution will bring about a better world, they may ask: ‘Better for whom?’ Related to both this and humility, is that an important distinction an intelligent person will be capable of drawing is that subjective opinion is totally separate from objective truth. Less intelligent individuals might be quite confident that their deeply held beliefs are unassailably true and correct, whereas an intelligent person is aware that the information they have is never complete, and will therefore never have the same conviction in what they say. This is otherwise known as the Dunning-Kruger effect.The Thousand Yard Stare: Many highly intelligent people are afflicted with what is known as 'low latent inhibition'; this describes the individual's relationship to the information their brain must process second-by-second. For less intelligent individuals, much of the information- sights, sounds, speech, and sensory data, is filtered away from the active analysis of the conscious mind. If the raw data outside does not resonate with established 'mental models', or ways of thinking about the world, the information is dismissed or quickly discarded. For more intelligent individuals however, mental models are more fluid; raw information is considered more actively and cogitation is a constant process. Again, more intelligent individuals place greater emphasis on deriving conclusions from the refinement of raw data rather than being told what their conclusion 'should' be. Behaviourally, this often expresses in an intelligent person as an intense and neutral stare, similar to those you might see in former soldiers with PTSD. What is occuring behind the scenes is a high degree of information processing, and subsequent distillation into a more accurate taxonomy of the reality outside.Peripheral: There's no getting around the fact that highly intelligent people do not represent the majority of the wider population; they are often at the periphery of social groups not only due to a difference in their manner of communication, which may be confusing in its complexity for less intelligent individuals, but also due to their inherent need for information consumption. If the intelligent person also has low latent inhibition, they may to a certain degree require a solitary nature to digest the vast current of information they are faced with each day. They might also recognise that time spent within social groups, while entertaining, might not broaden their knowledge base. While socialising, they are more likely to be drawn into gossip, spats or emotional situations; as the highly intelligent yearn for truth, scenarios involving subjectivity and feelings- even positive feelings- can be exhausting and seem counter-productive. During a group project, you will often find the highly intelligent students observing neutrally from a position at the edge of the group, ready to either lazily defer to the group's ideas, or to pounce into the conversation with particularly astute insights or plans.Delayed Gratification: In my time studying cognitive science, I was fascinated by how closely life success correlates to 'executive control'; this is the person's ability to direct their actions with consideration (there is a neural association with the prefrontal cortex). A classic study in executive control involves a group of child participants who are placed in a chair, with a single marshmallow left on the table in front of them, and are told that they can eat the single marshmallow now, or wait and receive a larger number later. Those children who were able to engage their executive control and wait for more were later observed to have achieved higher SAT scores, with fewer behavioural problems in a follow-up study. It is important here to emphasise that while executive control doesn’t have a direct relationship to IQ, as a factor in general intelligence it remains an important ability. [Edit: There are those in the comments that have taken issue with my inclusion of the marshmallow test, believing the study to have been discredited by later attempts at replication; I’ve done an audit of the major replication results and don’t think this is the right conclusion to make. While there are a number of other factors involved in the test, across the studies there remains a statistically significant effect from executive control that connects one’s ability to delay fulfilment to general cognitive ability. This connection has also been established in cuttlefish, chimpanzees and crows.]Incrementalism: When the highly intelligent are looking to reify a particular goal or set of circumstances that may be to the detriment of others, they will be aware of the necessity of managing adverse reactions. For example, if a factory owner concludes that in order for his factory to remain profitable it must cut wages significantly, a less intelligent means of enacting this wage cut would be to make the cut all at once. With employees facing such a significant change from one day to the next, the negative outcome is more salient. Employees may feel moved to strike, unionise, or otherwise take greater action to prevent the pay cuts. If however individual pay cuts are smaller, and made over longer periods of time, the negative changes undergo a process of ‘normalisation’ whereby they become socially acknowledged, if not accepted as ‘routine’ or mundane. The trajectory of pay change might trend quite negatively, but the average worker will rarely become aware of this, as regardless of the cuts, one day will seem much like the next. While there are potentially benign purposes for incrementalism, there is a far more prevalent, sinister power to it and the realities it is capable of creating; processes of long-term normalisation have, in the past, resulted in tryannical rule, genocide, massacre, and all manner of crude extremity. Those who wield this power of manipulation do indeed display high intelligence in their ability to foresee consequences and conduct long-term planning- but to speak for myself- I feel they often utilise it at the expense of their shared humanity and with a degree of personal cowardice.Esoteric: While it is possible to have a high IQ and not care for that abstract object we know as truth, it is more likely than not that the highly intelligent will invest themselves heavily in pursuit of it. For the intelligent person, they will have spent almost every moment of their lives in analysis; taking in the data and features of the world and arranging it so as to create coherence and predictive accuracy. This process of ‘model building’ is capable of tremendous leaps of insight, and the construction of links between disparate topics that places information in a new and more interesting light. This drive to become closer to ‘truth’, however, is always obstructed by our human limitations: we will never have all the data available to us, either through our personal faculties or through our technologies- as our perception is limited, so too is our ability to draw accurate conclusions. Taking this into account, the highly intelligent, in their unrelenting pursuit of truth, might be attracted to less conventional disciplines or areas of thought, if only to consider their potential to generate a more complete picture of things. There is a concept in the Greek Orthodox tradition, for example, known as ‘Henosis’, that describes the process whereby a human being is able to assimilate themselves with wider reality, thus deifying themselves and not only becoming one with ‘Truth’, but becoming ‘Truth’ itself. Similar transmutative descriptions have been made in the mystical offshoots of Christianity, Islam and Judaism, and are central to Western alchemy, Hermeticism, Buddhism and Taoism. Such concepts bear particular appeal to the highly intelligent as a means to arrive at something definitive: as they have sought truth industriously as long as they have been alive, the possibility of a path to understanding and awareness of all things, and an answer to all things, may be of existential importance to them.