Inner automorphism

Key takeaways

Inner automorphism belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Inner automorphism to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Inner automorphism from memory before moving on to harder problems.

Reference excerpt

In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.

Definition If G is a group and g is an element of G (alternatively, if G is a ring, and g is a unit), then the function

φ g : G → G φ g ( x ) := g − 1 x g {\displaystyle {\begin{aligned}\varphi _{g}\colon G&\to G\\\varphi _{g}(x)&:=g^{-1}xg\end{aligned}}}

is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all x 1 , x 2 ∈ G , {\displaystyle x_{1},x_{2}\in G,}

φ g ( x 1 x 2 ) = g − 1 x 1 x 2 g = g − 1 x 1 ( g g − 1 ) x 2 g = ( g − 1 x 1 g ) ( g − 1 x 2 g ) = φ g ( x 1 ) φ g ( x 2 ) , {\displaystyle \varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=g^{-1}x_{1}\left(gg^{-1}\right)x_{2}g=\left(g^{-1}x_{1}g\right)\left(g^{-1}x_{2}g\right)=\varphi _{g}(x_{1})\varphi _{g}(x_{2}),}

where the second equality is given by the insertion of the identity between x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} . Furthermore, it has a left and right inverse, namely φ g − 1 {\displaystyle \varphi _{g^{-1}}} . Thus, φ g {\displaystyle \varphi _{g}} is both a monomorphism and epimorphism, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

When discussing right conjugation, the expression g − 1 x g {\displaystyle g^{-1}xg} is often denoted exponentially by x g {\displaystyle x^{g}} . This notation is used because composition of conjugations satisfies the identity: ( x g 1 ) g 2 = x g 1 g 2 {\displaystyle \left(x^{g_{1}}\right)^{g_{2}}=x^{g_{1}g_{2}}} for all g 1 , g 2 ∈ G {\displaystyle g_{1},g_{2}\in G} . This shows that right conjugation gives a right action of G on itself. A common example is as follows:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Inner automorphism

Start with the simplest possible case. Write down what Inner automorphism claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Inner automorphism before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Inner automorphism ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Inner automorphism

In research: Inner automorphism appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Inner automorphism in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Inner automorphism is common in secondary-school and first-year university syllabi. It links to neighbouring topics Group automorphisms, Group theory, so understanding it makes those chapters shorter.
In everyday life: Look for Inner automorphism outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Inner automorphism in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Inner automorphism means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Inner automorphism out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Inner automorphism in simple terms?

In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via operations from within the group itself, hence the adjective "inner".

Why does Inner automorphism matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Inner automorphism?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Inner automorphism.