https://jinerx.github.io/tags/abstract-algebra/Recent content in Abstract Algebra on JinerX blogHugo -- gohugo.ioen-us© 2026 Jędrzej SajnógThu, 05 Mar 2026 17:25:33 +0100https://jinerx.github.io/learning_log/groups-abstract-algebra/Thu, 05 Mar 2026 17:25:33 +0100https://jinerx.github.io/learning_log/groups-abstract-algebra/<p>Here we consider definition of a group and some simple properties, operations, examples.</p> <h2 class="relative group">Fromal definition <div id="fromal-definition" class="anchor"></div> <span class="absolute top-0 w-6 transition-opacity opacity-0 -start-6 not-prose group-hover:opacity-100 select-none"> <a class="text-primary-300 dark:text-neutral-700 !no-underline" href="#fromal-definition" aria-label="Anchor">#</a> </span> </h2> <h3 class="relative group">Group <div id="group" class="anchor"></div> <span class="absolute top-0 w-6 transition-opacity opacity-0 -start-6 not-prose group-hover:opacity-100 select-none"> <a class="text-primary-300 dark:text-neutral-700 !no-underline" href="#group" aria-label="Anchor">#</a> </span> </h3> <p>A pair $(X, +)$ is called a group if $X$ is a set, $+$ is a function $+:X\times X\to X$ and:</p> <ul> <li>operations are associative: $a + (b + c) = (a + b) + c$</li> <li>X has an identity element 0 such that $\forall x\in X$ $x + 0 = 0 + x = x$</li> <li>each element $x$ has an inverse $x^{-1}$: $x+ x^{-1} = x^{-1}+ x = 0$</li> </ul> <h3 class="relative group">Semigrups <div id="semigrups" class="anchor"></div> <span class="absolute top-0 w-6 transition-opacity opacity-0 -start-6 not-prose group-hover:opacity-100 select-none"> <a class="text-primary-300 dark:text-neutral-700 !no-underline" href="#semigrups" aria-label="Anchor">#</a> </span> </h3> <p>Semigroup is a wekaer structure than the group it does not require:</p>